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Theorem elex2 2813
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 2365 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
21alrimiv 1621 . 2  |-  ( A  e.  B  ->  A. x
( x  =  A  ->  x  e.  B
) )
3 elisset 2811 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
4 exim 1565 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
52, 3, 4sylc 56 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696
This theorem is referenced by:  negn0  10320  nocvxmin  24416  itg2addnclem2  25004  risci  26721  dvh1dimat  32253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-v 2803
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