Theorem inn0 39244

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

Assertion

Ref	Expression
inn0	|- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )

Distinct variable groups:   x, A   x, B

Proof of Theorem inn0

Step	Hyp	Ref	Expression
1		nfcv 2764	. 2 |- F/_ x A
2		nfcv 2764	. 2 |- F/_ x B
3	1, 2	inn0f 39242	1 |- ( ( A i^i B ) =/= (/) <-> E. x e. A x e. B )

Syntax hints:   <-> wb 196   e. wcel 1990   =/= 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:  qinioo  39762