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Theorem opswap 5836
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 5828 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21unieqd 4640 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 opex 5122 . . . 4 𝐵, 𝐴⟩ ∈ V
43unisn 4646 . . 3 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
52, 4syl6eq 2856 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6 uni0 4659 . . 3 ∅ = ∅
7 opprc 4618 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
87sneqd 4382 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
98cnveqd 5499 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
10 cnvsn0 5814 . . . . 5 {∅} = ∅
119, 10syl6eq 2856 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
1211unieqd 4640 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
13 ancom 450 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14 opprc 4618 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
1513, 14sylnbi 321 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
166, 12, 153eqtr4a 2866 . 2 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
175, 16pm2.61i 176 1 {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  c0 4116  {csn 4370  cop 4376   cuni 4630  ccnv 5310
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 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
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 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-xp 5317  df-rel 5318  df-cnv 5319  df-dm 5321  df-rn 5322
This theorem is referenced by:  2nd1st  7445  cnvf1olem  7509  brtpos  7596  dftpos4  7606  tpostpos  7607  xpcomco  8289  fsumcnv  14727  fprodcnv  14934  gsumcom2  18575  txswaphmeolem  21821
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