Theorem elex22 2871

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 2422	. . . 4 |- (A e. B -> (x = A -> x e. B))
2		eleq1a 2422	. . . 4 |- (A e. C -> (x = A -> x e. C))
3	1, 2	anim12ii 553	. . 3 |- ((A e. B /\ A e. C) -> (x = A -> (x e. B /\ x e. C)))
4	3	alrimiv 1631	. 2 |- ((A e. B /\ A e. C) -> A.x(x = A -> (x e. B /\ x e. C)))
5		elisset 2870	. . 3 |- (A e. B -> E.x x = A)
6	5	adantr 451	. 2 |- ((A e. B /\ A e. C) -> E.x x = A)
7		exim 1575	. 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 358  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by: (None)