Theorem elex22 3217

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 2696	. . . 4 |- ( A e. B -> ( x = A -> x e. B ) )
2		eleq1a 2696	. . . 4 |- ( A e. C -> ( x = A -> x e. C ) )
3	1, 2	anim12ii 594	. . 3 |- ( ( A e. B /\ A e. C ) -> ( x = A -> ( x e. B /\ x e. C ) ) )
4	3	alrimiv 1855	. 2 |- ( ( A e. B /\ A e. C ) -> A. x ( x = A -> ( x e. B /\ x e. C ) ) )
5		elisset 3215	. . 3 |- ( A e. B -> E. x x = A )
6	5	adantr 481	. 2 |- ( ( A e. B /\ A e. C ) -> E. x x = A )
7		exim 1761	. 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 65	1 |- ( ( A e. B /\ A e. C ) -> E. x ( x e. B /\ x e. C ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990

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-12 2047  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-tru 1486  df-ex 1705  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202

This theorem is referenced by:  en3lplem1VD  39078