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Theorem cnvcnvsn 4827
 Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4833, this does not need any sethood assumptions on and .) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
cnvcnvsn

Proof of Theorem cnvcnvsn
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4733 . 2
2 relcnv 4733 . 2
3 vex 2605 . . . 4
4 vex 2605 . . . 4
53, 4opelcnv 4545 . . 3
6 ancom 262 . . . . . 6
73, 4opth 4000 . . . . . 6
84, 3opth 4000 . . . . . 6
96, 7, 83bitr4i 210 . . . . 5
103, 4opex 3992 . . . . . 6
1110elsn 3422 . . . . 5
124, 3opex 3992 . . . . . 6
1312elsn 3422 . . . . 5
149, 11, 133bitr4i 210 . . . 4
154, 3opelcnv 4545 . . . 4
163, 4opelcnv 4545 . . . 4
1714, 15, 163bitr4i 210 . . 3
185, 17bitri 182 . 2
191, 2, 18eqrelriiv 4460 1
 Colors of variables: wff set class Syntax hints:   wa 102   wceq 1285   wcel 1434  csn 3406  cop 3409  ccnv 4370 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:  rnsnopg  4829  cnvsn  4833
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