Theorem elex22 2563

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 2106	. . . 4 |- (A e. B -> (x = A -> x e. B))
2		eleq1a 2106	. . . 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 1751	. 2 |- ((A e. B /\ A e. C) -> A.x(x = A -> (x e. B /\ x e. C)))
5		elisset 2562	. . 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 1487	. 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 1240   = wceq 1242  E.wex 1378   e. wcel 1390

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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by: (None)