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Theorem cnvsng 5539
 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 4340 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21sneqd 4137 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
32cnveqd 5220 . . 3 (𝑥 = 𝐴{⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
4 opeq2 4341 . . . 4 (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
54sneqd 4137 . . 3 (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
63, 5eqeq12d 2625 . 2 (𝑥 = 𝐴 → ({⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ {⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7 opeq2 4341 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
87sneqd 4137 . . . 4 (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
98cnveqd 5220 . . 3 (𝑦 = 𝐵{⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
10 opeq1 4340 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
1110sneqd 4137 . . 3 (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
129, 11eqeq12d 2625 . 2 (𝑦 = 𝐵 → ({⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13 vex 3176 . . 3 𝑥 ∈ V
14 vex 3176 . . 3 𝑦 ∈ V
1513, 14cnvsn 5536 . 2 {⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
166, 12, 15vtocl2g 3243 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {csn 4125  ⟨cop 4131  ◡ccnv 5037 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046 This theorem is referenced by:  opswap  5540  funsng  5851  f1oprswap  6092  constr2spthlem1  26124  constr3pthlem2  26184
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