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Theorem cnvcnvsn 4740
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4746, 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 4646 . 2 Rel {⟨A, B⟩}
2 relcnv 4646 . 2 Rel {⟨B, A⟩}
3 vex 2554 . . . 4 y V
4 vex 2554 . . . 4 x V
53, 4opelcnv 4460 . . 3 (⟨y, x {⟨A, B⟩} ↔ ⟨x, y {⟨A, B⟩})
6 ancom 253 . . . . . 6 ((y = A x = B) ↔ (x = B y = A))
73, 4opth 3965 . . . . . 6 (⟨y, x⟩ = ⟨A, B⟩ ↔ (y = A x = B))
84, 3opth 3965 . . . . . 6 (⟨x, y⟩ = ⟨B, A⟩ ↔ (x = B y = A))
96, 7, 83bitr4i 201 . . . . 5 (⟨y, x⟩ = ⟨A, B⟩ ↔ ⟨x, y⟩ = ⟨B, A⟩)
103, 4opex 3957 . . . . . 6 y, x V
1110elsnc 3390 . . . . 5 (⟨y, x {⟨A, B⟩} ↔ ⟨y, x⟩ = ⟨A, B⟩)
124, 3opex 3957 . . . . . 6 x, y V
1312elsnc 3390 . . . . 5 (⟨x, y {⟨B, A⟩} ↔ ⟨x, y⟩ = ⟨B, A⟩)
149, 11, 133bitr4i 201 . . . 4 (⟨y, x {⟨A, B⟩} ↔ ⟨x, y {⟨B, A⟩})
154, 3opelcnv 4460 . . . 4 (⟨x, y {⟨A, B⟩} ↔ ⟨y, x {⟨A, B⟩})
163, 4opelcnv 4460 . . . 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 4377 1 {⟨A, B⟩} = {⟨B, A⟩}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {csn 3367  cop 3370  ccnv 4287
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  rnsnopg  4742  cnvsn  4746
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