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Theorem opswap 6185
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 6179 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21unieqd 4883 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 opex 5425 . . . 4 𝐵, 𝐴⟩ ∈ V
43unisn 4891 . . 3 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
52, 4eqtrdi 2789 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6 uni0 4900 . . 3 ∅ = ∅
7 opprc 4857 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
87sneqd 4602 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
98cnveqd 5835 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
10 cnvsn0 6166 . . . . 5 {∅} = ∅
119, 10eqtrdi 2789 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
1211unieqd 4883 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
13 ancom 462 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14 opprc 4857 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
1513, 14sylnbi 330 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
166, 12, 153eqtr4a 2799 . 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 397   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  {csn 4590  cop 4596   cuni 4869  ccnv 5636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644  df-cnv 5645  df-dm 5647  df-rn 5648
This theorem is referenced by:  2nd1st  7974  cnvf1olem  8046  brtpos  8170  dftpos4  8180  tpostpos  8181  xpcomco  9012  fsumcnv  15666  fprodcnv  15874  gsumcom2  19760  txswaphmeolem  23178
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