Theorem iunn0

Description: There is a nonempty class in an indexed collection B ( x ) iff the indexed union of them is nonempty. (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	iunn0	$$⊢\exists x\in AB\ne \varnothing \leftrightarrow \bigcup _{x\in A}B\ne \varnothing$$

Proof

Step	Hyp	Ref	Expression
1		rexcom4	$$⊢\exists x\in A\exists yy\in B\leftrightarrow \exists y\exists x\in Ay\in B$$
2		eliun	$$⊢y\in \bigcup _{x\in A}B\leftrightarrow \exists x\in Ay\in B$$
3	2	exbii	$$⊢\exists yy\in \bigcup _{x\in A}B\leftrightarrow \exists y\exists x\in Ay\in B$$
4	1 3	bitr4i	$$⊢\exists x\in A\exists yy\in B\leftrightarrow \exists yy\in \bigcup _{x\in A}B$$
5		n0	$$⊢B\ne \varnothing \leftrightarrow \exists yy\in B$$
6	5	rexbii	$$⊢\exists x\in AB\ne \varnothing \leftrightarrow \exists x\in A\exists yy\in B$$
7		n0	$$⊢\bigcup _{x\in A}B\ne \varnothing \leftrightarrow \exists yy\in \bigcup _{x\in A}B$$
8	4 6 7	3bitr4i	$$⊢\exists x\in AB\ne \varnothing \leftrightarrow \bigcup _{x\in A}B\ne \varnothing$$