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Theorem elex2 2547
Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.)
Assertion
Ref Expression
elex2 (A Bx x B)
Distinct variable groups:   x,A   x,B

Proof of Theorem elex2
StepHypRef Expression
1 eleq1a 2091 . . 3 (A B → (x = Ax B))
21alrimiv 1736 . 2 (A Bx(x = Ax B))
3 elisset 2545 . 2 (A Bx x = A)
4 exim 1472 . 2 (x(x = Ax B) → (x x = Ax x B))
52, 3, 4sylc 56 1 (A Bx x B)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1226   = wceq 1228  wex 1362   wcel 1374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1316  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-v 2537
This theorem is referenced by:  snmg  3460  oprcl  3547  exss  3937  regexmidlemm  4201  acexmidlem2  5433  gtso  6697
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