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Theorem ercl2 6856
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1  |-  ( ph  ->  R  Er  X )
ersym.2  |-  ( ph  ->  A R B )
Assertion
Ref Expression
ercl2  |-  ( ph  ->  B  e.  X )

Proof of Theorem ercl2
StepHypRef Expression
1 ersym.1 . 2  |-  ( ph  ->  R  Er  X )
2 ersym.2 . . 3  |-  ( ph  ->  A R B )
31, 2ersym 6855 . 2  |-  ( ph  ->  B R A )
41, 3ercl 6854 1  |-  ( ph  ->  B  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1717   class class class wbr 4155    Er wer 6840
This theorem is referenced by:  qliftfun  6927  efgcpbl2  15318  frgpcpbl  15320
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-br 4156  df-opab 4210  df-xp 4826  df-rel 4827  df-cnv 4828  df-dm 4830  df-er 6843
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