Theorem elex2 2872

Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)

Assertion

Ref	Expression
elex2	|- (A e. B -> E.x x e. B)

Distinct variable groups:   x,A   x,B

Proof of Theorem elex2

Step	Hyp	Ref	Expression
1		eleq1a 2422	. . 3 |- (A e. B -> (x = A -> x e. B))
2	1	alrimiv 1631	. 2 |- (A e. B -> A.x(x = A -> x e. B))
3		elisset 2870	. 2 |- (A e. B -> E.x x = A)
4		exim 1575	. 2 |- (A.x(x = A -> x e. B) -> (E.x x = A -> E.x x e. B))
5	2, 3, 4	sylc 56	1 |- (A e. B -> E.x x e. B)

Syntax hints:   -> wi 4  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)