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Theorem elex2 3125
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 2550 . . 3  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
21alrimiv 1695 . 2  |-  ( A  e.  B  ->  A. x
( x  =  A  ->  x  e.  B
) )
3 elisset 3124 . 2  |-  ( A  e.  B  ->  E. x  x  =  A )
4 exim 1633 . 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 setvar class
Syntax hints:    -> wi 4   A.wal 1377    = wceq 1379   E.wex 1596    e. wcel 1767
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-v 3115
This theorem is referenced by:  negn0  11169  nocvxmin  29304  itg2addnclem2  29920  risci  30220  dvh1dimat  36455
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