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Theorem erex 7354
 Description: An equivalence relation is a set if its domain is a set. (Contributed by Rodolfo Medina, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 12-Aug-2015.)
Assertion
Ref Expression
erex

Proof of Theorem erex
StepHypRef Expression
1 erssxp 7353 . . 3
2 sqxpexg 6605 . . 3
3 ssexg 4598 . . 3
41, 2, 3syl2an 477 . 2
54ex 434 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  e.wcel 1818   cvv 3109  C_wss 3475  X.cxp 5002  Erwer 7327 This theorem is referenced by:  erexb  7355  qliftlem  7411  qshash  13639  qusaddvallem  14948  qusaddflem  14949  qusaddval  14950  qusaddf  14951  qusmulval  14952  qusmulf  14953  qusgrp2  16188  efgrelexlemb  16768  efgcpbllemb  16773  frgpuplem  16790  qusring2  17269  vitalilem2  22018  vitalilem3  22019 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015  df-er 7330
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