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Theorem cnvsng 4733
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
Assertion
Ref Expression
cnvsng ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})

Proof of Theorem cnvsng
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 3523 . . . . 5 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
21sneqd 3363 . . . 4 (x = A → {⟨x, y⟩} = {⟨A, y⟩})
32cnveqd 4438 . . 3 (x = A{⟨x, y⟩} = {⟨A, y⟩})
4 opeq2 3524 . . . 4 (x = A → ⟨y, x⟩ = ⟨y, A⟩)
54sneqd 3363 . . 3 (x = A → {⟨y, x⟩} = {⟨y, A⟩})
63, 5eqeq12d 2036 . 2 (x = A → ({⟨x, y⟩} = {⟨y, x⟩} ↔ {⟨A, y⟩} = {⟨y, A⟩}))
7 opeq2 3524 . . . . 5 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
87sneqd 3363 . . . 4 (y = B → {⟨A, y⟩} = {⟨A, B⟩})
98cnveqd 4438 . . 3 (y = B{⟨A, y⟩} = {⟨A, B⟩})
10 opeq1 3523 . . . 4 (y = B → ⟨y, A⟩ = ⟨B, A⟩)
1110sneqd 3363 . . 3 (y = B → {⟨y, A⟩} = {⟨B, A⟩})
129, 11eqeq12d 2036 . 2 (y = B → ({⟨A, y⟩} = {⟨y, A⟩} ↔ {⟨A, B⟩} = {⟨B, A⟩}))
13 vex 2538 . . 3 x V
14 vex 2538 . . 3 y V
1513, 14cnvsn 4730 . 2 {⟨x, y⟩} = {⟨y, x⟩}
166, 12, 15vtocl2g 2594 1 ((A 𝑉 B 𝑊) → {⟨A, B⟩} = {⟨B, A⟩})
Colors of variables: wff set class
Syntax hints:  wi 4   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:  opswapg  4734  funsng  4872
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