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Theorem ercl2 6910
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 6909 . 2  |-  ( ph  ->  B R A )
41, 3ercl 6908 1  |-  ( ph  ->  B  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   class class class wbr 4204    Er wer 6894
This theorem is referenced by:  qliftfun  6981  efgcpbl2  15381  frgpcpbl  15383
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878  df-dm 4880  df-er 6897
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