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Theorem opswap 6251
Description: Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.)
Assertion
Ref Expression
opswap {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴

Proof of Theorem opswap
StepHypRef Expression
1 cnvsng 6245 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21unieqd 4925 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 opex 5475 . . . 4 𝐵, 𝐴⟩ ∈ V
43unisn 4931 . . 3 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
52, 4eqtrdi 2791 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6 uni0 4940 . . 3 ∅ = ∅
7 opprc 4901 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
87sneqd 4643 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
98cnveqd 5889 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
10 cnvsn0 6232 . . . . 5 {∅} = ∅
119, 10eqtrdi 2791 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
1211unieqd 4925 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
13 ancom 460 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14 opprc 4901 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
1513, 14sylnbi 330 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
166, 12, 153eqtr4a 2801 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
175, 16pm2.61i 182 1 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  cop 4637   cuni 4912  ccnv 5688
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  2nd1st  8062  cnvf1olem  8134  brtpos  8259  dftpos4  8269  tpostpos  8270  xpcomco  9101  fsumcnv  15806  fprodcnv  16016  gsumcom2  20008  txswaphmeolem  23828
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