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Theorem cnvcnvsn 5490
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5496, 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 5379 . 2
2 relcnv 5379 . 2
3 vex 3112 . . . 4
4 vex 3112 . . . 4
53, 4opelcnv 5189 . . 3
6 ancom 450 . . . . . 6
73, 4opth 4726 . . . . . 6
84, 3opth 4726 . . . . . 6
96, 7, 83bitr4i 277 . . . . 5
10 opex 4716 . . . . . 6
1110elsnc 4053 . . . . 5
12 opex 4716 . . . . . 6
1312elsnc 4053 . . . . 5
149, 11, 133bitr4i 277 . . . 4
154, 3opelcnv 5189 . . . 4
163, 4opelcnv 5189 . . . 4
1714, 15, 163bitr4i 277 . . 3
185, 17bitri 249 . 2
191, 2, 18eqrelriiv 5102 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  {csn 4029  <.cop 4035  `'ccnv 5003
This theorem is referenced by:  rnsnopg  5492  cnvsn  5496  strlemor1  14724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  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 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012
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