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Theorem cnvsngOLD 5764
 Description: Obsolete proof of cnvsng 5757 as of 12-Feb-2022. (Contributed by NM, 23-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cnvsngOLD ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})

Proof of Theorem cnvsngOLD
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opeq1 4540 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21sneqd 4329 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
32cnveqd 5435 . . 3 (𝑥 = 𝐴{⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
4 opeq2 4541 . . . 4 (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
54sneqd 4329 . . 3 (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
63, 5eqeq12d 2786 . 2 (𝑥 = 𝐴 → ({⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ {⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7 opeq2 4541 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
87sneqd 4329 . . . 4 (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
98cnveqd 5435 . . 3 (𝑦 = 𝐵{⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
10 opeq1 4540 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
1110sneqd 4329 . . 3 (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
129, 11eqeq12d 2786 . 2 (𝑦 = 𝐵 → ({⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13 vex 3354 . . 3 𝑥 ∈ V
14 vex 3354 . . 3 𝑦 ∈ V
1513, 14cnvsn 5760 . 2 {⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
166, 12, 15vtocl2g 3421 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  {csn 4317  ⟨cop 4323  ◡ccnv 5249 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-cnv 5258 This theorem is referenced by: (None)
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