Theorem elex22 2569

Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
elex22	|- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem elex22

Step	Hyp	Ref	Expression
1		eleq1a 2109	. . . 4 |- ( A e. B -> ( x = A -> x e. B ) )
2		eleq1a 2109	. . . 4 |- ( A e. C -> ( x = A -> x e. C ) )
3	1, 2	anim12ii 325	. . 3 |- ( ( A e. B /\ A e. C ) -> ( x = A -> ( x e. B /\ x e. C ) ) )
4	3	alrimiv 1754	. 2 |- ( ( A e. B /\ A e. C ) -> A. x ( x = A -> ( x e. B /\ x e. C ) ) )
5		elisset 2568	. . 3 |- ( A e. B -> E. x x = A )
6	5	adantr 261	. 2 |- ( ( A e. B /\ A e. C ) -> E. x x = A )
7		exim 1490	. 2 |- ( A. x ( x = A -> ( x e. B /\ x e. C ) ) -> ( E. x x = A -> E. x ( x e. B /\ x e. C ) ) )
8	4, 6, 7	sylc 56	1 |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   = wceq 1243   E.wex 1381   e. wcel 1393

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-v 2559

This theorem is referenced by: (None)