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Theorem ercl2 6576
Description: Elementhood in the field of an equivalence relation. (Contributed by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ercl2 (𝜑𝐵𝑋)

Proof of Theorem ercl2
StepHypRef Expression
1 ersym.1 . 2 (𝜑𝑅 Er 𝑋)
2 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
31, 2ersym 6575 . 2 (𝜑𝐵𝑅𝐴)
41, 3ercl 6574 1 (𝜑𝐵𝑋)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2160   class class class wbr 4021   Er wer 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-dm 4657  df-er 6563
This theorem is referenced by:  qliftfun  6647  nqnq0pi  7472  qusgrp2  13078
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