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Theorem cnvsng 4856
Description: Converse of a singleton of an ordered pair. (Contributed by NM, 23-Jan-2015.)
Assertion
Ref Expression
cnvsng ((𝐴𝑉𝐵𝑊) → {⟨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 3590 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21sneqd 3429 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
32cnveqd 4559 . . 3 (𝑥 = 𝐴{⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
4 opeq2 3591 . . . 4 (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
54sneqd 3429 . . 3 (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
63, 5eqeq12d 2097 . 2 (𝑥 = 𝐴 → ({⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ {⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7 opeq2 3591 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
87sneqd 3429 . . . 4 (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
98cnveqd 4559 . . 3 (𝑦 = 𝐵{⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
10 opeq1 3590 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
1110sneqd 3429 . . 3 (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
129, 11eqeq12d 2097 . 2 (𝑦 = 𝐵 → ({⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13 vex 2613 . . 3 𝑥 ∈ V
14 vex 2613 . . 3 𝑦 ∈ V
1513, 14cnvsn 4853 . 2 {⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
166, 12, 15vtocl2g 2671 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434  {csn 3416  cop 3419  ccnv 4390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399
This theorem is referenced by:  opswapg  4857  funsng  4996
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