Theorem elex22 2767

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 2261	. . . 4 |- ( A e. B -> ( x = A -> x e. B ) )
2		eleq1a 2261	. . . 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 1885	. 2 |- ( ( A e. B /\ A e. C ) -> A. x ( x = A -> ( x e. B /\ x e. C ) ) )
5		elisset 2766	. . 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 1610	. 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 1362   = wceq 1364   E.wex 1503   e. wcel 2160

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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754

This theorem is referenced by: (None)