Theorem inn0f 39242

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

Hypotheses

Ref	Expression
inn0f.1	|- F/_ x A
inn0f.2	|- F/_ x B

Assertion

Ref	Expression
inn0f	|- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )

Proof of Theorem inn0f

Step	Hyp	Ref	Expression
1		elin 3796	. . 3 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
2	1	exbii 1774	. 2 |- ( E. x x e. ( A i^i B ) <-> E. x ( x e. A /\ x e. B ) )
3		inn0f.1	. . . 4 |- F/_ x A
4		inn0f.2	. . . 4 |- F/_ x B
5	3, 4	nfin 3820	. . 3 |- F/_ x ( A i^i B )
6	5	n0f 3927	. 2 |- ( ( A i^i B ) =/= (/) <-> E. x x e. ( A i^i B ) )
7		df-rex 2918	. 2 |- ( E. x e. A x e. B <-> E. x ( x e. A /\ x e. B ) )
8	2, 6, 7	3bitr4i 292	1 |- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   F/_wnfc 2751   =/= wne 2794   E.wrex 2913   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-in 3581  df-nul 3916

This theorem is referenced by:  inn0  39244