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Theorem cnvsym 4738
 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 1408 . 2
2 relcnv 4733 . . 3
3 ssrel 4454 . . 3
42, 3ax-mp 7 . 2
5 vex 2605 . . . . . 6
6 vex 2605 . . . . . 6
75, 6brcnv 4546 . . . . 5
8 df-br 3794 . . . . 5
97, 8bitr3i 184 . . . 4
10 df-br 3794 . . . 4
119, 10imbi12i 237 . . 3
12112albii 1401 . 2
131, 4, 123bitr4i 210 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 103  wal 1283   wcel 1434   wss 2974  cop 3409   class class class wbr 3793  ccnv 4370   wrel 4376 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379 This theorem is referenced by:  dfer2  6173
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