ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ercl2 Unicode version

Theorem ercl2 6495
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 6494 . 2  |-  ( ph  ->  B R A )
41, 3ercl 6493 1  |-  ( ph  ->  B  e.  X )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2128   class class class wbr 3967    Er wer 6479
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-br 3968  df-opab 4028  df-xp 4594  df-rel 4595  df-cnv 4596  df-dm 4598  df-er 6482
This theorem is referenced by:  qliftfun  6564  nqnq0pi  7360
  Copyright terms: Public domain W3C validator