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Theorem ersym 7706
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (𝜑𝑅 Er 𝑋)
ersym.2 (𝜑𝐴𝑅𝐵)
Assertion
Ref Expression
ersym (𝜑𝐵𝑅𝐴)

Proof of Theorem ersym
StepHypRef Expression
1 ersym.2 . . 3 (𝜑𝐴𝑅𝐵)
2 ersym.1 . . . . . 6 (𝜑𝑅 Er 𝑋)
3 errel 7703 . . . . . 6 (𝑅 Er 𝑋 → Rel 𝑅)
42, 3syl 17 . . . . 5 (𝜑 → Rel 𝑅)
5 brrelex12 5120 . . . . 5 ((Rel 𝑅𝐴𝑅𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
64, 1, 5syl2anc 692 . . . 4 (𝜑 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
7 brcnvg 5268 . . . . 5 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵𝑅𝐴𝐴𝑅𝐵))
87ancoms 469 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵𝑅𝐴𝐴𝑅𝐵))
96, 8syl 17 . . 3 (𝜑 → (𝐵𝑅𝐴𝐴𝑅𝐵))
101, 9mpbird 247 . 2 (𝜑𝐵𝑅𝐴)
11 df-er 7694 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 ∧ dom 𝑅 = 𝑋 ∧ (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
1211simp3bi 1076 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
132, 12syl 17 . . . 4 (𝜑 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
1413unssad 3773 . . 3 (𝜑𝑅𝑅)
1514ssbrd 4661 . 2 (𝜑 → (𝐵𝑅𝐴𝐵𝑅𝐴))
1610, 15mpd 15 1 (𝜑𝐵𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3189  cun 3557  wss 3559   class class class wbr 4618  ccnv 5078  dom cdm 5079  ccom 5083  Rel wrel 5084   Er wer 7691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-xp 5085  df-rel 5086  df-cnv 5087  df-er 7694
This theorem is referenced by:  ercl2  7707  ersymb  7708  ertr2d  7711  ertr3d  7712  ertr4d  7713  erth  7743  erinxp  7773  nqereu  9703  nqerf  9704  1nqenq  9736  qusgrp2  17465  efginvrel2  18072  efgcpbllemb  18100  2idlcpbl  19166  tgptsmscls  21876
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