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Theorem cnvsng 4882
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 3605 . . . . 5 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
21sneqd 3444 . . . 4 (𝑥 = 𝐴 → {⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
32cnveqd 4580 . . 3 (𝑥 = 𝐴{⟨𝑥, 𝑦⟩} = {⟨𝐴, 𝑦⟩})
4 opeq2 3606 . . . 4 (𝑥 = 𝐴 → ⟨𝑦, 𝑥⟩ = ⟨𝑦, 𝐴⟩)
54sneqd 3444 . . 3 (𝑥 = 𝐴 → {⟨𝑦, 𝑥⟩} = {⟨𝑦, 𝐴⟩})
63, 5eqeq12d 2099 . 2 (𝑥 = 𝐴 → ({⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩} ↔ {⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩}))
7 opeq2 3606 . . . . 5 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
87sneqd 3444 . . . 4 (𝑦 = 𝐵 → {⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
98cnveqd 4580 . . 3 (𝑦 = 𝐵{⟨𝐴, 𝑦⟩} = {⟨𝐴, 𝐵⟩})
10 opeq1 3605 . . . 4 (𝑦 = 𝐵 → ⟨𝑦, 𝐴⟩ = ⟨𝐵, 𝐴⟩)
1110sneqd 3444 . . 3 (𝑦 = 𝐵 → {⟨𝑦, 𝐴⟩} = {⟨𝐵, 𝐴⟩})
129, 11eqeq12d 2099 . 2 (𝑦 = 𝐵 → ({⟨𝐴, 𝑦⟩} = {⟨𝑦, 𝐴⟩} ↔ {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}))
13 vex 2618 . . 3 𝑥 ∈ V
14 vex 2618 . . 3 𝑦 ∈ V
1513, 14cnvsn 4879 . 2 {⟨𝑥, 𝑦⟩} = {⟨𝑦, 𝑥⟩}
166, 12, 15vtocl2g 2676 1 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1287  wcel 1436  {csn 3431  cop 3434  ccnv 4410
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4417  df-rel 4418  df-cnv 4419
This theorem is referenced by:  opswapg  4883  funsng  5025
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