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Theorem cnvcnvsn 5831
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5838, 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 5720 . 2 Rel {⟨𝐴, 𝐵⟩}
2 relcnv 5720 . 2 Rel {⟨𝐵, 𝐴⟩}
3 vex 3401 . . . 4 𝑥 ∈ V
4 vex 3401 . . . 4 𝑦 ∈ V
53, 4opelcnv 5512 . . 3 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩})
6 ancom 450 . . . . . 6 ((𝑥 = 𝐴𝑦 = 𝐵) ↔ (𝑦 = 𝐵𝑥 = 𝐴))
73, 4opth 5141 . . . . . 6 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴𝑦 = 𝐵))
84, 3opth 5141 . . . . . 6 (⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩ ↔ (𝑦 = 𝐵𝑥 = 𝐴))
96, 7, 83bitr4i 294 . . . . 5 (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
10 opex 5129 . . . . . 6 𝑥, 𝑦⟩ ∈ V
1110elsn 4392 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
12 opex 5129 . . . . . 6 𝑦, 𝑥⟩ ∈ V
1312elsn 4392 . . . . 5 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐵, 𝐴⟩)
149, 11, 133bitr4i 294 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
154, 3opelcnv 5512 . . . 4 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩})
163, 4opelcnv 5512 . . . 4 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐵, 𝐴⟩})
1714, 15, 163bitr4i 294 . . 3 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
185, 17bitri 266 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐵, 𝐴⟩})
191, 2, 18eqrelriiv 5423 1 {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1637  wcel 2157  {csn 4377  cop 4383  ccnv 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-br 4852  df-opab 4914  df-xp 5324  df-rel 5325  df-cnv 5326
This theorem is referenced by:  rnsnopg  5833  cnvsng  5835  cnvsnOLD  5839  strlemor1OLD  16183
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