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Theorem elex2 2564
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 2106 . . 3 (A B → (x = Ax B))
21alrimiv 1751 . 2 (A Bx(x = Ax B))
3 elisset 2562 . 2 (A Bx x = A)
4 exim 1487 . 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 1240   = wceq 1242  wex 1378   wcel 1390
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 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553
This theorem is referenced by:  snmg  3477  oprcl  3564  exss  3954  regexmidlemm  4217  acexmidlem2  5452  enm  6230  ssfiexmid  6254  gtso  6874  indstr  8292  negm  8306  fzm  8652  fzom  8770
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