Theorem eluni

Description: Membership in class union. (Contributed by NM, 22-May-1994)

	Ref	Expression
Assertion	eluni	$$⊢A\in \bigcup B\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)$$

Proof

Step	Hyp	Ref	Expression
1		elex	$$⊢A\in \bigcup B\to A\in \mathrm{V}$$
2		elex	$$⊢A\in x\to A\in \mathrm{V}$$
3	2	adantr	$$⊢\left(A\in x\wedge x\in B\right)\to A\in \mathrm{V}$$
4	3	exlimiv	$$⊢\exists x\left(A\in x\wedge x\in B\right)\to A\in \mathrm{V}$$
5		eleq1	$$⊢y=A\to \left(y\in x\leftrightarrow A\in x\right)$$
6	5	anbi1d	$$⊢y=A\to \left(\left(y\in x\wedge x\in B\right)\leftrightarrow \left(A\in x\wedge x\in B\right)\right)$$
7	6	exbidv	$$⊢y=A\to \left(\exists x\left(y\in x\wedge x\in B\right)\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)\right)$$
8		df-uni	$$⊢\bigcup B=\left\{y|\exists x\left(y\in x\wedge x\in B\right)\right\}$$
9	7 8	elab2g	$$⊢A\in \mathrm{V}\to \left(A\in \bigcup B\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)\right)$$
10	1 4 9	pm5.21nii	$$⊢A\in \bigcup B\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)$$