Theorem elex2 2746

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 2242	. . 3 |- ( A e. B -> ( x = A -> x e. B ) )
2	1	alrimiv 1867	. 2 |- ( A e. B -> A. x ( x = A -> x e. B ) )
3		elisset 2744	. 2 |- ( A e. B -> E. x x = A )
4		exim 1592	. 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 1346   = wceq 1348   E.wex 1485   e. wcel 2141

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  snmg  3701  oprcl  3789  brm  4039  ss1o0el1  4183  exss  4212  onintrab2im  4502  regexmidlemm  4516  dmxpid  4832  acexmidlem2  5850  frecabcl  6378  ixpm  6708  enm  6798  ssfilem  6853  fin0  6863  fin0or  6864  diffitest  6865  diffisn  6871  infm  6882  inffiexmid  6884  ctssdc  7090  omct  7094  ctssexmid  7126  exmidfodomrlemr  7179  exmidfodomrlemrALT  7180  exmidaclem  7185  caucvgsrlemasr  7752  suplocsrlempr  7769  gtso  7998  sup3exmid  8873  indstr  9552  negm  9574  fzm  9994  fzom  10120  rexfiuz  10953  r19.2uz  10957  resqrexlemgt0  10984  climuni  11256  bezoutlembi  11960  lcmgcdlem  12031  pcprecl  12243  pc2dvds  12283  nninfdclemcl  12403  tgioo  13340  pw1nct  14036  nninfall  14042