Theorem eluni2

Description: Membership in class union. Restricted quantifier version. (Contributed by NM, 31-Aug-1999)

	Ref	Expression
Assertion	eluni2	$$⊢A\in \bigcup B\leftrightarrow \exists x\in BA\in x$$

Proof

Step	Hyp	Ref	Expression
1		exancom	$$⊢\exists x\left(A\in x\wedge x\in B\right)\leftrightarrow \exists x\left(x\in B\wedge A\in x\right)$$
2		eluni	$$⊢A\in \bigcup B\leftrightarrow \exists x\left(A\in x\wedge x\in B\right)$$
3		df-rex	$$⊢\exists x\in BA\in x\leftrightarrow \exists x\left(x\in B\wedge A\in x\right)$$
4	1 2 3	3bitr4i	$$⊢A\in \bigcup B\leftrightarrow \exists x\in BA\in x$$