Theorem inn0f

Description: A nonempty intersection. (Contributed by Glauco Siliprandi, 24-Dec-2020)

	Ref	Expression
Hypotheses	inn0f.1	|- F/_ x A
	inn0f.2	|- F/_ x B
Assertion	inn0f	|- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )

Proof

Step	Hyp	Ref	Expression
1		inn0f.1	|- F/_ x A
2		inn0f.2	|- F/_ x B
3		elin	|- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
4	3	exbii	|- ( E. x x e. ( A i^i B ) <-> E. x ( x e. A /\ x e. B ) )
5	1 2	nfin	|- F/_ x ( A i^i B )
6	5	n0f	|- ( ( A i^i B ) =/= (/) <-> E. x x e. ( A i^i B ) )
7		df-rex	|- ( E. x e. A x e. B <-> E. x ( x e. A /\ x e. B ) )
8	4 6 7	3bitr4i	|- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )