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Theorem erex 8287
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 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))

Proof of Theorem erex
StepHypRef Expression
1 erssxp 8286 . . 3 (𝑅 Er 𝐴𝑅 ⊆ (𝐴 × 𝐴))
2 sqxpexg 7451 . . 3 (𝐴𝑉 → (𝐴 × 𝐴) ∈ V)
3 ssexg 5199 . . 3 ((𝑅 ⊆ (𝐴 × 𝐴) ∧ (𝐴 × 𝐴) ∈ V) → 𝑅 ∈ V)
41, 2, 3syl2an 597 . 2 ((𝑅 Er 𝐴𝐴𝑉) → 𝑅 ∈ V)
54ex 415 1 (𝑅 Er 𝐴 → (𝐴𝑉𝑅 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  Vcvv 3470  wss 3909   × cxp 5525   Er wer 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-xp 5533  df-rel 5534  df-cnv 5535  df-dm 5537  df-rn 5538  df-er 8263
This theorem is referenced by:  erexb  8288  qliftlem  8352  qshash  15158  qusaddvallem  16799  qusaddflem  16800  qusaddval  16801  qusaddf  16802  qusmulval  16803  qusmulf  16804  qusgrp2  18192  efgrelexlemb  18851  efgcpbllemb  18856  frgpuplem  18873  qusring2  19345  vitalilem2  24188  vitalilem3  24189  tgjustr  26243
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