Theorem 0pssin 38064

Description: Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)

Assertion

Ref	Expression
0pssin	|- ( (/) C. ( A i^i B ) <-> E. x ( x e. A /\ x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem 0pssin

Step	Hyp	Ref	Expression
1		0pss 4013	. 2 |- ( (/) C. ( A i^i B ) <-> ( A i^i B ) =/= (/) )
2		ndisj 38063	. 2 |- ( ( A i^i B ) =/= (/) <-> E. x ( x e. A /\ x e. B ) )
3	1, 2	bitri 264	1 |- ( (/) C. ( A i^i B ) <-> E. x ( x e. A /\ x e. B ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   e. wcel 1990   =/= wne 2794   i^i cin 3573   C. wpss 3575   (/)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-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916

This theorem is referenced by: (None)