Theorem elex2 2741

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 2237	. . 3 |- ( A e. B -> ( x = A -> x e. B ) )
2	1	alrimiv 1862	. 2 |- ( A e. B -> A. x ( x = A -> x e. B ) )
3		elisset 2739	. 2 |- ( A e. B -> E. x x = A )
4		exim 1587	. 2 |- ( A. x ( x = A -> x e. B ) -> ( E. x x = A -> E. x x e. B ) )
5	2, 3, 4	sylc 62	1 |- ( A e. B -> E. x x e. B )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-v 2727

This theorem is referenced by:  snmg  3693  oprcl  3781  brm  4031  ss1o0el1  4175  exss  4204  onintrab2im  4494  regexmidlemm  4508  dmxpid  4824  acexmidlem2  5838  frecabcl  6363  ixpm  6692  enm  6782  ssfilem  6837  fin0  6847  fin0or  6848  diffitest  6849  diffisn  6855  infm  6866  inffiexmid  6868  ctssdc  7074  omct  7078  ctssexmid  7110  exmidfodomrlemr  7154  exmidfodomrlemrALT  7155  exmidaclem  7160  caucvgsrlemasr  7727  suplocsrlempr  7744  gtso  7973  sup3exmid  8848  indstr  9527  negm  9549  fzm  9969  fzom  10095  rexfiuz  10927  r19.2uz  10931  resqrexlemgt0  10958  climuni  11230  bezoutlembi  11934  lcmgcdlem  12005  pcprecl  12217  pc2dvds  12257  nninfdclemcl  12377  tgioo  13146  pw1nct  13843  nninfall  13849