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Theorem cnvcnvsn 4724
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4730, this does not need any sethood assumptions on A and B.) (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
cnvcnvsn {⟨A, B⟩} = {⟨B, A⟩}

Proof of Theorem cnvcnvsn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 4630 . 2 Rel {⟨A, B⟩}
2 relcnv 4630 . 2 Rel {⟨B, A⟩}
3 vex 2538 . . . 4 y V
4 vex 2538 . . . 4 x V
53, 4opelcnv 4444 . . 3 (⟨y, x {⟨A, B⟩} ↔ ⟨x, y {⟨A, B⟩})
6 ancom 253 . . . . . 6 ((y = A x = B) ↔ (x = B y = A))
73, 4opth 3948 . . . . . 6 (⟨y, x⟩ = ⟨A, B⟩ ↔ (y = A x = B))
84, 3opth 3948 . . . . . 6 (⟨x, y⟩ = ⟨B, A⟩ ↔ (x = B y = A))
96, 7, 83bitr4i 201 . . . . 5 (⟨y, x⟩ = ⟨A, B⟩ ↔ ⟨x, y⟩ = ⟨B, A⟩)
103, 4opex 3940 . . . . . 6 y, x V
1110elsnc 3373 . . . . 5 (⟨y, x {⟨A, B⟩} ↔ ⟨y, x⟩ = ⟨A, B⟩)
124, 3opex 3940 . . . . . 6 x, y V
1312elsnc 3373 . . . . 5 (⟨x, y {⟨B, A⟩} ↔ ⟨x, y⟩ = ⟨B, A⟩)
149, 11, 133bitr4i 201 . . . 4 (⟨y, x {⟨A, B⟩} ↔ ⟨x, y {⟨B, A⟩})
154, 3opelcnv 4444 . . . 4 (⟨x, y {⟨A, B⟩} ↔ ⟨y, x {⟨A, B⟩})
163, 4opelcnv 4444 . . . 4 (⟨y, x {⟨B, A⟩} ↔ ⟨x, y {⟨B, A⟩})
1714, 15, 163bitr4i 201 . . 3 (⟨x, y {⟨A, B⟩} ↔ ⟨y, x {⟨B, A⟩})
185, 17bitri 173 . 2 (⟨y, x {⟨A, B⟩} ↔ ⟨y, x {⟨B, A⟩})
191, 2, 18eqrelriiv 4361 1 {⟨A, B⟩} = {⟨B, A⟩}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1228   wcel 1374  {csn 3350  cop 3353  ccnv 4271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280
This theorem is referenced by:  rnsnopg  4726  cnvsn  4730
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