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Theorem elex2 3044
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 2544 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
21alrimiv 1781 . 2  |-  ( A  e.  B  ->  A. x
( x  =  A  ->  x  e.  B
) )
3 elisset 3043 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
4 exim 1714 . 2  |-  ( A. x ( x  =  A  ->  x  e.  B )  ->  ( E. x  x  =  A  ->  E. x  x  e.  B ) )
52, 3, 4sylc 61 1  |-  ( A  e.  B  ->  E. x  x  e.  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-12 1950  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-v 3033
This theorem is referenced by:  negn0  10069  nocvxmin  30651  itg2addnclem2  32058  risci  32290  dvh1dimat  35080
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