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Theorem elex2 2587
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
StepHypRef Expression
1 eleq1a 2125 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
21alrimiv 1770 . 2  |-  ( A  e.  B  ->  A. x
( x  =  A  ->  x  e.  B
) )
3 elisset 2585 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
4 exim 1506 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
52, 3, 4sylc 60 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  snmg  3514  oprcl  3601  exss  3991  onintrab2im  4272  regexmidlemm  4285  acexmidlem2  5537  enm  6325  ssfiexmid  6367  fin0  6373  fin0or  6374  diffitest  6375  diffisn  6381  caucvgsrlemasr  6932  gtso  7156  indstr  8632  negm  8647  fzm  9004  fzom  9122  rexfiuz  9816  r19.2uz  9820  resqrexlemgt0  9847  climuni  10045
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