Theorem elex2 3216

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 2696	. . 3 |- ( A e. B -> ( x = A -> x e. B ) )
2	1	alrimiv 1855	. 2 |- ( A e. B -> A. x ( x = A -> x e. B ) )
3		elisset 3215	. 2 |- ( A e. B -> E. x x = A )
4		exim 1761	. 2 |- ( A. x ( x = A -> x e. B ) -> ( E. x x = A -> E. x x e. B ) )
5	2, 3, 4	sylc 65	1 |- ( A e. B -> E. x x e. B )

Syntax hints:   -> wi 4   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:  negn0  10459  nocvxmin  31894  itg2addnclem2  33462  risci  33786  dvh1dimat  36730