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Theorem elex22 2648
Description: If two classes each contain another class, then both contain some set. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
elex22  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Distinct variable groups:    x, A    x, B    x, C

Proof of Theorem elex22
StepHypRef Expression
1 eleq1a 2166 . . . 4  |-  ( A  e.  B  ->  (
x  =  A  ->  x  e.  B )
)
2 eleq1a 2166 . . . 4  |-  ( A  e.  C  ->  (
x  =  A  ->  x  e.  C )
)
31, 2anim12ii 336 . . 3  |-  ( ( A  e.  B  /\  A  e.  C )  ->  ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) ) )
43alrimiv 1809 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  A. x ( x  =  A  ->  (
x  e.  B  /\  x  e.  C )
) )
5 elisset 2647 . . 3  |-  ( A  e.  B  ->  E. x  x  =  A )
65adantr 271 . 2  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x  x  =  A )
7 exim 1542 . 2  |-  ( A. x ( x  =  A  ->  ( x  e.  B  /\  x  e.  C ) )  -> 
( E. x  x  =  A  ->  E. x
( x  e.  B  /\  x  e.  C
) ) )
84, 6, 7sylc 62 1  |-  ( ( A  e.  B  /\  A  e.  C )  ->  E. x ( x  e.  B  /\  x  e.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1294    = wceq 1296   E.wex 1433    e. wcel 1445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1388  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-v 2635
This theorem is referenced by: (None)
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