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Theorem cnvsym 5057
 Description: Two ways of saying a relation is symmetric. Similar to definition of symmetry in [Schechter] p. 51. (Contributed by NM, 28-Dec-1996.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvsym
Distinct variable group:   ,,

Proof of Theorem cnvsym
StepHypRef Expression
1 alcom 1711 . 2
2 relcnv 5051 . . 3
3 ssrel 4776 . . 3
42, 3ax-mp 8 . 2
5 vex 2791 . . . . . 6
6 vex 2791 . . . . . 6
75, 6brcnv 4864 . . . . 5
8 df-br 4024 . . . . 5
97, 8bitr3i 242 . . . 4
10 df-br 4024 . . . 4
119, 10imbi12i 316 . . 3
12112albii 1554 . 2
131, 4, 123bitr4i 268 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176  wal 1527   wcel 1684   wss 3152  cop 3643   class class class wbr 4023  ccnv 4688   wrel 4694 This theorem is referenced by:  dfer2  6661  twsymr  25078 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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