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Theorem cnvsng 5036
 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 3715 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21sneqd 3547 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
32cnveqd 4727 . . 3 (𝑥 = 𝐴{⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
4 opeq2 3716 . . . 4 (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
54sneqd 3547 . . 3 (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
63, 5eqeq12d 2156 . 2 (𝑥 = 𝐴 → ({⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ {⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7 opeq2 3716 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
87sneqd 3547 . . . 4 (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
98cnveqd 4727 . . 3 (𝑦 = 𝐵{⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
10 opeq1 3715 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
1110sneqd 3547 . . 3 (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
129, 11eqeq12d 2156 . 2 (𝑦 = 𝐵 → ({⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13 vex 2694 . . 3 𝑥 ∈ V
14 vex 2694 . . 3 𝑦 ∈ V
1513, 14cnvsn 5033 . 2 {⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
166, 12, 15vtocl2g 2755 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  {csn 3534  ⟨cop 3537  ◡ccnv 4550 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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2123  ax-sep 4056  ax-pow 4108  ax-pr 4142 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1738  df-eu 2004  df-mo 2005  df-clab 2128  df-cleq 2134  df-clel 2137  df-nfc 2272  df-ral 2423  df-rex 2424  df-v 2693  df-un 3082  df-in 3084  df-ss 3091  df-pw 3519  df-sn 3540  df-pr 3541  df-op 3543  df-br 3940  df-opab 4000  df-xp 4557  df-rel 4558  df-cnv 4559 This theorem is referenced by:  opswapg  5037  funsng  5181
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