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Theorem cnvcnvsn 5485
 Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5491, 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 5374 . 2
2 relcnv 5374 . 2
3 vex 3116 . . . 4
4 vex 3116 . . . 4
53, 4opelcnv 5184 . . 3
6 ancom 450 . . . . . 6
73, 4opth 4721 . . . . . 6
84, 3opth 4721 . . . . . 6
96, 7, 83bitr4i 277 . . . . 5
10 opex 4711 . . . . . 6
1110elsnc 4051 . . . . 5
12 opex 4711 . . . . . 6
1312elsnc 4051 . . . . 5
149, 11, 133bitr4i 277 . . . 4
154, 3opelcnv 5184 . . . 4
163, 4opelcnv 5184 . . . 4
1714, 15, 163bitr4i 277 . . 3
185, 17bitri 249 . 2
191, 2, 18eqrelriiv 5097 1
 Colors of variables: wff setvar class Syntax hints:   wa 369   wceq 1379   wcel 1767  csn 4027  cop 4033  ccnv 4998 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007 This theorem is referenced by:  rnsnopg  5487  cnvsn  5491  strlemor1  14582
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