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Theorem cnvsym 5391
 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 1846 . 2
2 relcnv 5384 . . 3
3 ssrel 5100 . . 3
42, 3ax-mp 5 . 2
5 vex 3112 . . . . . 6
6 vex 3112 . . . . . 6
75, 6brcnv 5195 . . . . 5
8 df-br 4457 . . . . 5
97, 8bitr3i 251 . . . 4
10 df-br 4457 . . . 4
119, 10imbi12i 326 . . 3
12112albii 1642 . 2
131, 4, 123bitr4i 277 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184  wal 1393   wcel 1819   wss 3471  cop 4038   class class class wbr 4456  ccnv 5007   wrel 5013 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016 This theorem is referenced by:  dfer2  7330
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