# 并集

## 有限聯集

${\displaystyle (\forall A)(\forall B)(\forall x)\left\{(x\in A\cup B)\Leftrightarrow \left[(x\in A)\vee (x\in B)\right]\right\}}$

「對所有 ${\displaystyle x}$${\displaystyle x\in A\cup B}$ 等價於 ${\displaystyle x\in A}$${\displaystyle x\in B}$

${\displaystyle x}$${\displaystyle A\cup B\cup C}$的元素，当且仅当${\displaystyle x}$属于${\displaystyle A}$${\displaystyle x}$属于${\displaystyle B}$${\displaystyle x}$属于${\displaystyle C}$

### 代数性质

${\displaystyle A\cup (B\cup C)=(A\cup B)\cup C}$。事实上，${\displaystyle A\cup B\cup C}$也等于这两个集合，因此圆括号在仅进行并集运算的时候可以省略。

## 无限并集

${\displaystyle (\forall {\mathcal {M}})(\forall x)\left\{\left(x\in \bigcup {\mathcal {M}}\right)\Leftrightarrow (\exists A)\left[\left(A\in {\mathcal {M}}\right)\wedge (x\in A)\right]\right\}}$

${\displaystyle X\subseteq \bigcup {\mathcal {M}}}$ ，會稱 ${\displaystyle X}$${\displaystyle {\mathcal {M}}}$ 覆蓋（cover），也就是直觀上可以用 ${\displaystyle {\mathcal {M}}}$ 裡的所有集合疊起來蓋住 ${\displaystyle X}$

${\displaystyle {\mathcal {M}}=\{A,B,C\}}$${\displaystyle \bigcup {\mathcal {M}}=A\cup B\cup C}$ ，若 ${\displaystyle M}$空集${\displaystyle \bigcup {\mathcal {M}}}$ 也是空集。

${\displaystyle \bigcup _{A\in {\mathcal {M}}}A}$

${\displaystyle I\,{\overset {A}{\cong }}\,{\mathcal {M}}}$${\displaystyle \bigcup _{i\in I}A(i):=\bigcup {\mathcal {M}}}$

${\displaystyle \mathbb {N} \,{\overset {A}{\cong }}\,{\mathcal {M}}}$${\displaystyle \bigcup _{i=0}^{\infty }A(i):=\bigcup {\mathcal {M}}}$

### 无限并集的性質

${\displaystyle \vdash \bigcup \varnothing =\varnothing }$

(1) ${\displaystyle (\forall x)[\neg (x\in \varnothing )]}$ (空集公理)

(2) ${\displaystyle \neg (S\in \varnothing )}$(MP with A4, 1)

(3)${\displaystyle (S\in \varnothing )\Rightarrow [\neg (x\in S)]}$(M0 with 2)

(4)${\displaystyle \neg \neg (S\in \varnothing )\Rightarrow [\neg (x\in S)]}$(Equv with DN, 3)

(5)${\displaystyle \neg \{[\neg (S\in \varnothing )]\wedge [\neg (x\in S)]\}}$(Equv with De Morgan, 4)

(6)${\displaystyle (\forall S){\big \{}\neg \{[\neg (S\in \varnothing )]\wedge [\neg (x\in S)]\}{\big \}}}$(GEN with ${\displaystyle S}$ , 5)

(7)${\displaystyle \neg (\exists S)\{[\neg (S\in \varnothing )]\wedge [\neg (x\in S)]\}}$(Equv with DN, 6)

(8)${\displaystyle (\forall x)\left\{\left(x\in \bigcup \varnothing \right)\Leftrightarrow (\exists S)\{(S\in \varnothing )\wedge (x\in S)\}\right\}}$(MP with 并集公理, A4)

(9)${\displaystyle \left(x\in \bigcup \varnothing \right)\Leftrightarrow (\exists S)\{(S\in \varnothing )\wedge (x\in S)\}}$(MP with A4, 8)

(10)${\displaystyle \left(x\in \bigcup \varnothing \right)\Rightarrow (\exists S)\{(S\in \varnothing )\wedge (x\in S)\}}$(MP with AND ,9)

(11)${\displaystyle \neg (\exists S)\{(S\in \varnothing )\wedge (x\in S)\}\Rightarrow \neg \left(x\in \bigcup \varnothing \right)}$(MP with T, 10)

(12)${\displaystyle \neg \left(x\in \bigcup \varnothing \right)}$(MP with 7, 11)

(13)${\displaystyle (\forall x)\left(x\not \in \bigcup \varnothing \right)}$(GEN with ${\displaystyle x}$ , 12)

(14)${\displaystyle (y=\varnothing )\Leftrightarrow (\forall x)[\neg (x\in y)]}$ (E)

(15)${\displaystyle (\forall y)\{(y=\varnothing )\Leftrightarrow (\forall x)[\neg (x\in y)]\}}$ (GEN with ${\displaystyle y}$ , 14)

(16)${\displaystyle \left(\bigcup \varnothing =\varnothing \right)\Leftrightarrow (\forall x)\left[\neg \left(x\in \bigcup \varnothing \right)\right]}$(MP with A4, 15)

(17) ${\displaystyle \bigcup \varnothing =\varnothing }$ (Equv with 13, 16)

#### 比較性質

${\displaystyle ({\mathcal {M}}\subseteq {\mathcal {N}})\vdash \left(\bigcup {\mathcal {M}}\subseteq \bigcup {\mathcal {N}}\right)}$

${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}}),\,{\mathcal {P}}\vdash {\mathcal {Q}}\wedge {\mathcal {R}}}$

(u) ${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}})\vdash {\mathcal {P}}\Rightarrow ({\mathcal {Q}}\wedge {\mathcal {R}})}$

(1) ${\displaystyle (\forall A)\left[(A\in {\mathcal {M}})\Rightarrow (A\in {\mathcal {N}})\right]}$ (Hyp)

(2) ${\displaystyle (A\in {\mathcal {M}})\Rightarrow (A\in {\mathcal {N}})}$(MP with 1, A4)

(3) ${\displaystyle [(a\in A)\wedge (A\in {\mathcal {M}})]\Rightarrow (A\in {\mathcal {M}})}$(AND)

(4)${\displaystyle [(a\in A)\wedge (A\in {\mathcal {M}})]\Rightarrow (a\in A)}$(AND)

(5)${\displaystyle [(a\in A)\wedge (A\in {\mathcal {M}})]\Rightarrow (A\in {\mathcal {N}})}$(D1 with 2, 3)

(6)${\displaystyle [(a\in A)\wedge (A\in {\mathcal {M}})]\Rightarrow [(a\in A)\wedge (A\in {\mathcal {N}})]}$(u with 4, 5)

(7)${\displaystyle (\exists A\in {\mathcal {M}})(a\in A)\Rightarrow (\exists A\in {\mathcal {N}})(a\in A)}$(GENe with ${\displaystyle A}$, 6)

(8) ${\displaystyle (\forall x)\left\{\left(x\in \bigcup {\mathcal {M}}\right)\Leftrightarrow (\exists A\in {\mathcal {M}})(x\in A)\right\}}$(MP with 并集公理, A4)

(9) ${\displaystyle (\forall x)\left\{\left(x\in \bigcup {\mathcal {N}}\right)\Leftrightarrow (\exists A\in {\mathcal {N}})(x\in A)\right\}}$(MP with 并集公理, A4)

(10) ${\displaystyle \left(x\in \bigcup {\mathcal {M}}\right)\Leftrightarrow (\exists A\in {\mathcal {M}})(x\in A)}$ (MP with 8, A4)

(11) ${\displaystyle \left(x\in \bigcup {\mathcal {N}}\right)\Leftrightarrow (\exists A\in {\mathcal {N}})(x\in A)}$ (MP with 9, A4)

(12) ${\displaystyle \left(x\in \bigcup {\mathcal {M}}\right)\Rightarrow (\exists A\in {\mathcal {N}})(x\in A)}$(D1 with 7, 10)

(13) ${\displaystyle \left(x\in \bigcup {\mathcal {M}}\right)\Rightarrow \left(x\in \bigcup {\mathcal {N}}\right)}$(D1 with 11, 12)

(14) ${\displaystyle (\forall x)\left[\left(x\in \bigcup {\mathcal {M}}\right)\Rightarrow \left(x\in \bigcup {\mathcal {N}}\right)\right]}$(GEN with ${\displaystyle a}$ , 13)

#### 覆蓋性質

${\displaystyle \vdash A=\bigcup {\mathcal {P}}(A)}$

${\displaystyle A}$ 正好就是其冪集的聯集」，這個定理直觀上可理解成，因為冪集 ${\displaystyle {\mathcal {P}}(A)}$ 是以 ${\displaystyle A}$${\displaystyle A}$子集為元素，所以 ${\displaystyle {\mathcal {P}}(A)}$ 的聯集理當是 ${\displaystyle A}$

${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}}),\,{\mathcal {P}}\vdash {\mathcal {Q}}\wedge {\mathcal {R}}}$

(u) ${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}})\vdash {\mathcal {P}}\Rightarrow ({\mathcal {Q}}\wedge {\mathcal {R}})}$

(1)${\displaystyle (\forall x)\left\{\left[x\in \bigcup {\mathcal {P}}(A)\right]\Leftrightarrow (\exists S)\{[S\in {\mathcal {P}}(A)]\wedge (x\in S)\}\right\}}$(MP with 并集公理, A4)

(2) ${\displaystyle (\forall S)\{[S\in {\mathcal {P}}(A)]\Leftrightarrow (S\subseteq A)\}}$(幂集公理)

(3) ${\displaystyle [S\in {\mathcal {P}}(A)]\Leftrightarrow (S\subseteq A)}$(MP with A4 ,2)

(4) ${\displaystyle (\forall x)\left\{\left[x\in \bigcup {\mathcal {P}}(A)\right]\Leftrightarrow (\exists S)[(S\subseteq A)\wedge (x\in S)]\right\}}$ (Equv with 1, 3)

(5) ${\displaystyle [(S\subseteq A)\wedge (x\in S)]\Rightarrow (S\subseteq A)}$(AND)

(6) ${\displaystyle (\forall x)[(x\in S)\Rightarrow (x\in A)]\Rightarrow [(x\in S)\Rightarrow (x\in A)]}$(A4)

(7) ${\displaystyle [(S\subseteq A)\wedge (x\in S)]\Rightarrow [(x\in S)\Rightarrow (x\in A)]}$(D1 with 5, 6)

(8) ${\displaystyle [(S\subseteq A)\wedge (x\in S)]\Rightarrow (x\in S)}$(AND)

(9) ${\displaystyle [(S\subseteq A)\wedge (x\in S)]\Rightarrow \{(x\in S)\wedge [(x\in S)\Rightarrow (x\in A)]\}}$(u with 7, 8)

${\displaystyle (x\in S),\,[(x\in S)\Rightarrow (x\in A)]\vdash (x\in A)}$

${\displaystyle \{(x\in S)\wedge [(x\in S)\Rightarrow (x\in A)]\}\vdash (x\in A)}$(a)

(10') ${\displaystyle [(S\subseteq A)\wedge (x\in S)]\Rightarrow (x\in A)}$(D1 with a, 9)

(11') ${\displaystyle (\exists S)[(S\subseteq A)\wedge (x\in S)]\Rightarrow (x\in A)}$(GENe with ${\displaystyle S}$, 10')

(12') ${\displaystyle (\forall S)\{\neg [(S\subseteq A)\wedge (x\in S)]\}\Rightarrow \{\neg [(A\subseteq A)\wedge (x\in A)]\}}$ (A4)

(13') ${\displaystyle [(A\subseteq A)\wedge (x\in A)]\Rightarrow (\exists S)[(S\subseteq A)\wedge (x\in S)]}$ (MP with T, 12')

(14') ${\displaystyle (x\in A)\Rightarrow (x\in A)}$ (I)

(15') ${\displaystyle A\subseteq A}$ (GEN with ${\displaystyle x}$ , 14')

${\displaystyle (A\subseteq A)\vdash (x\in A)\Rightarrow [(A\subseteq A)\wedge (x\in A)]}$(b)

(16'') ${\displaystyle (x\in A)\Rightarrow [(A\subseteq A)\wedge (x\in A)]}$ (b with 15')

(17'') ${\displaystyle (x\in A)\Rightarrow (\exists S)[(S\subseteq A)\wedge (x\in S)]}$ (D1 with 13', 16'')

(18'') ${\displaystyle (x\in A)\Leftrightarrow (\exists S)[(S\subseteq A)\wedge (x\in S)]}$ (AND with 11', 17'')

(19'') ${\displaystyle (\forall x)\left\{\left[x\in \bigcup {\mathcal {P}}(A)\right]\Leftrightarrow (x\in A)\right\}}$(Equv with 4, 18''')

${\displaystyle \left(\bigcup {\mathcal {M}}\subseteq A\right)\vdash (\forall M\in {\mathcal {M}})(M\subseteq A)}$

(1) ${\displaystyle (\forall a)\left[(\exists M\in {\mathcal {M}})(a\in M)\Rightarrow (a\in A)\right]}$ (Hyp)

(2) ${\displaystyle [(M\in {\mathcal {M}})\wedge (a\in M)]\Rightarrow (\exists M\in {\mathcal {M}})(a\in M)}$ (A4 and T)

(3) ${\displaystyle (\exists M\in {\mathcal {M}})(a\in M)\Rightarrow (a\in A)}$ (MP with 1, A4)

(4) ${\displaystyle [(M\in {\mathcal {M}})\wedge (a\in M)]\Rightarrow (a\in A)}$ (D1 with 2, 3)

(5) ${\displaystyle (M\in {\mathcal {M}})\Rightarrow [(a\in M)\Rightarrow (a\in A)]}$ (MP with abb, 4)

(6) ${\displaystyle (\forall a)\{(M\in {\mathcal {M}})\Rightarrow [(a\in M)\Rightarrow (a\in A)]\}}$ (GEN with ${\displaystyle a}$ , 5)

(7) ${\displaystyle (M\in {\mathcal {M}})\Rightarrow (\forall a)[(a\in M)\Rightarrow (a\in A)]}$ (MP with A5 , 6)

(8) ${\displaystyle (\forall M)\{(M\in {\mathcal {M}})\Rightarrow (\forall a)[(a\in M)\Rightarrow (a\in A)]\}}$ (GEN with ${\displaystyle M}$ , 7)

${\displaystyle \vdash \left(A=\bigcup {\mathcal {M}}\right)\Rightarrow \left\{A\subseteq \bigcup {\mathcal {[}}{\mathcal {M}}\cup {\mathcal {P}}(A)]\right\}}$

${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}}),\,{\mathcal {P}}\vdash {\mathcal {Q}}\wedge {\mathcal {R}}}$

(u) ${\displaystyle ({\mathcal {P}}\Rightarrow {\mathcal {Q}}),\,({\mathcal {P}}\Rightarrow {\mathcal {R}})\vdash {\mathcal {P}}\Rightarrow ({\mathcal {Q}}\wedge {\mathcal {R}})}$

(1) ${\displaystyle (\forall a)\left[(a\in A)\Leftrightarrow (\exists M\in {\mathcal {M}})(a\in M)\right]}$ (Hyp)

(2) ${\displaystyle (\forall M\in {\mathcal {M}})(M\subseteq A)}$(MP with 1, 定理3)

(3) ${\displaystyle (M\in {\mathcal {M}})\Rightarrow (M\subseteq A)}$(MP with A4, 2)

(4) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow (M\in {\mathcal {M}})}$(AND)

(5) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow (a\in M)}$(AND)

(6) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow (M\in {\mathcal {M}})}$(AND)

(7) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow (M\subseteq A)}$ (D1 with 3, 4)

(8) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow [(a\in M)\wedge (M\in {\mathcal {M}})]}$(a with 5, 6)

(9) ${\displaystyle [(a\in M)\wedge (M\in {\mathcal {M}})]\Rightarrow [(a\in M)\wedge (M\in {\mathcal {M}})\wedge (M\subseteq A)]}$(a with 7, 8)

(10) ${\displaystyle (\exists M\in {\mathcal {M}})(a\in M)\Rightarrow (\exists M\in {\mathcal {M}})[(a\in M)\wedge (M\subseteq A)]}$(GENe with ${\displaystyle M}$, 9)

(11) ${\displaystyle (a\in A)\Leftrightarrow (\exists M\in {\mathcal {M}})(a\in M)}$(MP with A4, 1)

(12) ${\displaystyle (a\in A)\Rightarrow (\exists M\in {\mathcal {M}})(a\in M)}$(AND with 11)

(13) ${\displaystyle (a\in A)\Rightarrow (\exists M\in {\mathcal {M}})[(a\in M)\wedge (M\subseteq A)]}$(D1 with 10, 12)

(14) ${\displaystyle (\forall a\in A)(\exists M\in {\mathcal {M}})[(a\in M)\wedge (M\subseteq A)]}$(GEN with ${\displaystyle a}$, 13)

(15)${\displaystyle (\forall S)\{[S\in {\mathcal {P}}(A)]\Leftrightarrow (S\subseteq A)\}}$(幂集公理)

(16)${\displaystyle [M\in {\mathcal {P}}(A)]\Leftrightarrow (M\subseteq A)}$(MP with A4, 15)

(17)${\displaystyle (\forall a\in A)(\exists M\in {\mathcal {M}})\{(a\in M)\wedge [M\in {\mathcal {P}}(A)]\}}$(Equv with 14, 16)

(18) ${\displaystyle (\forall A)(\forall B)(\forall x)\left\{(x\in A\cap B)\Leftrightarrow \left[(x\in A)\wedge (x\in B)\right]\right\}}$(有限交集)

(19)${\displaystyle (\forall B)(\forall x)\left\{(x\in {\mathcal {M}}\cap B)\Leftrightarrow \left[(x\in {\mathcal {M}})\wedge (x\in B)\right]\right\}}$(MP with A4, 18)

(20)${\displaystyle (\forall x){\big \{}[x\in {\mathcal {M}}\cap {\mathcal {P}}(A)]\Leftrightarrow \left\{(x\in {\mathcal {M}})\wedge [x\in {\mathcal {P}}(A)]\right\}{\big \}}}$(MP with A4, 19)

(21)${\displaystyle [M\in {\mathcal {M}}\cap {\mathcal {P}}(A)]\Leftrightarrow \left\{(M\in {\mathcal {M}})\wedge [M\in {\mathcal {P}}(A)]\right\}}$(MP with A4, 20)

(22)${\displaystyle (\forall a\in A)(\exists M)\{(a\in M)\wedge [M\in {\mathcal {M}}\cap {\mathcal {P}}(A)]\}}$(Equv with 17, 21)

(23)${\displaystyle \left\{a\in \bigcup [{\mathcal {M}}\cap {\mathcal {P}}(A)]\right\}\Leftrightarrow (\exists M)\{(a\in M)\wedge [M\in {\mathcal {M}}\cap {\mathcal {P}}(A)]\}}$(MP with 并集公理, A4)

(24)${\displaystyle (\forall a)\left\{(a\in A)\Rightarrow \left\{a\in \bigcup [{\mathcal {M}}\cap {\mathcal {P}}(A)]\right\}\right\}}$(Equv with 22, 23)

#### 運算性質

${\displaystyle {\mathcal {M}}_{A}:=\left\{B\,|\,(\exists M\in {\mathcal {M}})(B=M\cap A)\right\}}$

${\displaystyle \vdash \bigcup {\mathcal {M}}_{A}=A\cap \left(\bigcup {\mathcal {M}}\right)}$

(1)${\displaystyle (\forall B)[(B\in {\mathcal {M}}_{A})\Leftrightarrow (\exists M\in {\mathcal {M}})(B=M\cap A)]}$ (${\displaystyle {\mathcal {M}}_{A}}$的定義)

(2) ${\displaystyle (\forall x)\left\{\left(x\in \bigcup {\mathcal {M}}\right)\Leftrightarrow (\exists B\in {\mathcal {M}})(x\in B)\right\}}$(MP with 并集公理, A4)

(3) ${\displaystyle (\forall A)(\forall B)(\forall x)\left\{(x\in A\cap B)\Leftrightarrow \left[(x\in A)\wedge (x\in B)\right]\right\}}$(有限交集)

(4)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists B)[(B\in {\mathcal {M}}_{A})\wedge (x\in B)]}$(MP with A4, 2)

(5) ${\displaystyle (B\in {\mathcal {M}}_{A})\Leftrightarrow (\exists M\in {\mathcal {M}})(B=M\cap A)}$(MP with A4, 1)

(6) ${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists B)[(x\in B)\wedge (\exists M\in {\mathcal {M}})(B=M\cap A)]}$(Equv with 4, 5)

(7)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists B)(\exists M)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(Equv with Ce, 6)

(8)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists M)(\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(Equv with 量詞可交換性 ,7)

(9) ${\displaystyle (B=M\cap A)\Rightarrow \{[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\}}$(E2)

(10)${\displaystyle [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow (B=M\cap A)}$(AND)

(11)${\displaystyle [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$${\displaystyle \Rightarrow \{[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\}}$(D1 with 9,10)

(12)${\displaystyle \{[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\}}$

${\displaystyle \Rightarrow \{[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\}}$(MP with A2, 11)

(13)${\displaystyle [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(I)

(14)${\displaystyle [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]}$(MP with 12, 13)

(15)${\displaystyle [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})]}$(AND)

(16)${\displaystyle [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})]}$(D1 with 14,15)

(17)${\displaystyle (\exists M)(\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Rightarrow (\exists M)[(x\in M\cap A)\wedge (M\in {\mathcal {M}})]}$(GENe with ${\displaystyle B}$ then ${\displaystyle M}$)

(18)${\displaystyle M\cap A=M\cap A}$ (E1)

${\displaystyle {\mathcal {P}}\vdash {\mathcal {R}}\Rightarrow ({\mathcal {R}}\wedge {\mathcal {P}})}$(a)

(19)${\displaystyle [(x\in M\cap A)\wedge (M\in {\mathcal {M}})]\Rightarrow [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]}$(a with 18)

(20)${\displaystyle (\forall B)\{\neg [(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\}\Rightarrow \{\neg [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\}}$(A4)

(21)${\displaystyle [(x\in M\cap A)\wedge (M\in {\mathcal {M}})\wedge (M\cap A=M\cap A)]\Rightarrow (\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(MP with T, 20)

(22)${\displaystyle [(x\in M\cap A)\wedge (M\in {\mathcal {M}})]\Rightarrow (\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(D1 with 19, 21)

(23)${\displaystyle (\exists M)[(x\in M\cap A)\wedge (M\in {\mathcal {M}})]\Rightarrow (\exists M)(\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]}$(GENe with ${\displaystyle M}$)

(24)${\displaystyle (\exists M)(\exists B)[(x\in B)\wedge (M\in {\mathcal {M}})\wedge (B=M\cap A)]\Leftrightarrow (\exists M)[(x\in M\cap A)\wedge (M\in {\mathcal {M}})]}$(AND with 17, 23)

(25)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists M)[(x\in M\cap A)\wedge (M\in {\mathcal {M}})]}$(Equv with 8, 24)

(26) ${\displaystyle (x\in M\cap A)\Leftrightarrow [(x\in M)\wedge (x\in A)]}$(MP with A4, 3)

(27)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow (\exists M)[(x\in M)\wedge (x\in A)\wedge (M\in {\mathcal {M}})]}$(Equv with 25, 26)

(28)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow \{(\exists M)[(x\in M)\wedge (M\in {\mathcal {M}})]\wedge (x\in A)\}}$(Equv with Ce, 27)

(30) ${\displaystyle \left(x\in \bigcup {\mathcal {M}}\right)\Leftrightarrow (\exists M\in {\mathcal {M}})(x\in M)}$(MP with A4, 2)

(31)${\displaystyle \left(x\in \bigcup {\mathcal {M}}_{A}\right)\Leftrightarrow \left[\left(x\in \bigcup {\mathcal {M}}\right)\wedge (x\in A)\right]}$(Equv with 28, 30)

(32)${\displaystyle \left[x\in A\cap \left(\bigcup {\mathcal {M}}\right)\right]\Leftrightarrow \left[(x\in A)\wedge \left(x\in \bigcup {\mathcal {M}}\right)\right]}$(MP with A4, 3)

(33)${\displaystyle \left[x\in A\cap \left(\bigcup {\mathcal {M}}\right)\right]\Leftrightarrow \left(x\in \bigcup {\mathcal {M}}_{A}\right)}$(Equv with 31, 32)

(34)${\displaystyle (\forall x)\left\{\left[x\in A\cap \left(\bigcup {\mathcal {M}}\right)\right]\Leftrightarrow \left(x\in \bigcup {\mathcal {M}}_{A}\right)\right\}}$(GEN with ${\displaystyle x}$, 33)

${\displaystyle \bigcup _{i\in I}\left(A\cap B_{i}\right)=A\cap \bigcup _{i\in I}B_{i}}$

${\displaystyle \mathbb {N} \,{\overset {A}{\cong }}\,{\mathcal {A}}}$，若對自然数 ${\displaystyle m\in \mathbb {N} }$ 做以下的符號定義：

${\displaystyle {\mathcal {A}}_{m}:=\left\{S\in {\mathcal {A}}\,|\,A^{-1}(S)\geq m\right\}}$
${\displaystyle {\mathcal {I}}:=\left\{S\,{\bigg |}\,(\exists m\in \mathbb {N} )\left(S=\bigcap {\mathcal {A}}_{m}\right)\right\}}$
${\displaystyle {\mathcal {S}}:=\left\{S\,{\bigg |}\,(\exists m\in \mathbb {N} )\left(S=\bigcup {\mathcal {A}}_{m}\right)\right\}}$

${\displaystyle \vdash \bigcup {\mathcal {I}}\subseteq \bigcap {\mathcal {S}}}$

${\displaystyle \bigcup _{i=0}^{\infty }\left(\bigcap _{j=i}^{\infty }A_{j}\right)\subseteq \bigcap _{i=0}^{\infty }\left(\bigcup _{j=i}^{\infty }A_{j}\right)}$

## 参考文献

1. ^ 程极泰. 集合论. 应用数学丛书 第一版. 国防工业出版社. 1985: 14. 15034.2766.