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Theorem ercl2 6526
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 6525 . 2  |-  ( ph  ->  B R A )
41, 3ercl 6524 1  |-  ( ph  ->  B  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2141   class class class wbr 3989    Er wer 6510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621  df-er 6513
This theorem is referenced by:  qliftfun  6595  nqnq0pi  7400
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