Theorem elex2 2615

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 2150	. . 3 |- ( A e. B -> ( x = A -> x e. B ) )
2	1	alrimiv 1795	. 2 |- ( A e. B -> A. x ( x = A -> x e. B ) )
3		elisset 2613	. 2 |- ( A e. B -> E. x x = A )
4		exim 1530	. 2 |- ( A. x ( x = A -> x e. B ) -> ( E. x x = A -> E. x x e. B ) )
5	2, 3, 4	sylc 61	1 |- ( A e. B -> E. x x e. B )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by:  snmg  3508  oprcl  3594  exss  3982  onintrab2im  4262  regexmidlemm  4275  acexmidlem2  5529  enm  6317  ssfilem  6360  fin0  6369  fin0or  6370  diffitest  6371  diffisn  6377  caucvgsrlemasr  6966  gtso  7190  indstr  8681  negm  8700  fzm  9057  fzom  9173  rexfiuz  9875  r19.2uz  9879  resqrexlemgt0  9906  climuni  10132  bezoutlembi  10394  lcmgcdlem  10459