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Theorem cnvcnvsn 4983
 Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4989, 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 4885 . 2
2 relcnv 4885 . 2
3 vex 2661 . . . 4
4 vex 2661 . . . 4
53, 4opelcnv 4689 . . 3
6 ancom 264 . . . . . 6
73, 4opth 4127 . . . . . 6
84, 3opth 4127 . . . . . 6
96, 7, 83bitr4i 211 . . . . 5
103, 4opex 4119 . . . . . 6
1110elsn 3511 . . . . 5
124, 3opex 4119 . . . . . 6
1312elsn 3511 . . . . 5
149, 11, 133bitr4i 211 . . . 4
154, 3opelcnv 4689 . . . 4
163, 4opelcnv 4689 . . . 4
1714, 15, 163bitr4i 211 . . 3
185, 17bitri 183 . 2
191, 2, 18eqrelriiv 4601 1
 Colors of variables: wff set class Syntax hints:   wa 103   wceq 1314   wcel 1463  csn 3495  cop 3498  ccnv 4506 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-rel 4514  df-cnv 4515 This theorem is referenced by:  rnsnopg  4985  cnvsn  4989
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