Theorem elex22 2787

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 2277	. . . 4 |- ( A e. B -> ( x = A -> x e. B ) )
2		eleq1a 2277	. . . 4 |- ( A e. C -> ( x = A -> x e. C ) )
3	1, 2	anim12ii 343	. . 3 |- ( ( A e. B /\ A e. C ) -> ( x = A -> ( x e. B /\ x e. C ) ) )
4	3	alrimiv 1897	. 2 |- ( ( A e. B /\ A e. C ) -> A. x ( x = A -> ( x e. B /\ x e. C ) ) )
5		elisset 2786	. . 3 |- ( A e. B -> E. x x = A )
6	5	adantr 276	. 2 |- ( ( A e. B /\ A e. C ) -> E. x x = A )
7		exim 1622	. 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 62	1 |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774

This theorem is referenced by: (None)