Theorem uniex2

Description: The Axiom of Union using the standard abbreviation for union. Given any set x , its union y exists. (Contributed by NM, 4-Jun-2006) (Proof shortened by BJ, 14-Jul-2026)

	Ref	Expression
Assertion	uniex2	$$⊢\exists yy=\bigcup x$$

Proof

Step	Hyp	Ref	Expression
1		axun2	$$⊢\exists y\forall z\left(z\in y\leftrightarrow \exists w\left(z\in w\wedge w\in x\right)\right)$$
2		eluni	$$⊢z\in \bigcup x\leftrightarrow \exists w\left(z\in w\wedge w\in x\right)$$
3	2	bibi2i	$$⊢\left(z\in y\leftrightarrow z\in \bigcup x\right)\leftrightarrow \left(z\in y\leftrightarrow \exists w\left(z\in w\wedge w\in x\right)\right)$$
4	3	albii	$$⊢\forall z\left(z\in y\leftrightarrow z\in \bigcup x\right)\leftrightarrow \forall z\left(z\in y\leftrightarrow \exists w\left(z\in w\wedge w\in x\right)\right)$$
5	4	exbii	$$⊢\exists y\forall z\left(z\in y\leftrightarrow z\in \bigcup x\right)\leftrightarrow \exists y\forall z\left(z\in y\leftrightarrow \exists w\left(z\in w\wedge w\in x\right)\right)$$
6	1 5	mpbir	$$⊢\exists y\forall z\left(z\in y\leftrightarrow z\in \bigcup x\right)$$
7		dfcleq	$$⊢y=\bigcup x\leftrightarrow \forall z\left(z\in y\leftrightarrow z\in \bigcup x\right)$$
8	7	biimpri	$$⊢\forall z\left(z\in y\leftrightarrow z\in \bigcup x\right)\to y=\bigcup x$$
9	6 8	eximii	$$⊢\exists yy=\bigcup x$$