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Theorem opswap 5620
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 5619 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
21unieqd 4444 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
3 opex 4930 . . . 4 𝐵, 𝐴⟩ ∈ V
43unisn 4449 . . 3 {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴
52, 4syl6eq 2671 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6 uni0 4463 . . 3 ∅ = ∅
7 opprc 4422 . . . . . . 7 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
87sneqd 4187 . . . . . 6 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
98cnveqd 5296 . . . . 5 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
10 cnvsn0 5601 . . . . 5 {∅} = ∅
119, 10syl6eq 2671 . . . 4 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
1211unieqd 4444 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = ∅)
13 ancom 466 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14 opprc 4422 . . . 4 (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
1513, 14sylnbi 320 . . 3 (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
166, 12, 153eqtr4a 2681 . 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 1482  wcel 1989  Vcvv 3198  c0 3913  {csn 4175  cop 4181   cuni 4434  ccnv 5111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123
This theorem is referenced by:  2nd1st  7210  cnvf1olem  7272  brtpos  7358  dftpos4  7368  tpostpos  7369  xpcomco  8047  fsumcnv  14498  fprodcnv  14707  gsumcom2  18368  txswaphmeolem  21601
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