Theorem opswap 6231

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

Step	Hyp	Ref	Expression
1		cnvsng 6225	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4889	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5446	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4895	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2820	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4905	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4865	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4606	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5862	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ◡{∅})
10		cnvsn0 6212	. . . . 5 ⊢ ◡{∅} = ∅
11	9, 10	eqtrdi 2820	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4889	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 465	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4865	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 333	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2830	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
17	5, 16	pm2.61i 184	1 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4594  ⟨cop 4600  ∪ cuni 4876  ^◡ccnv 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673

This theorem is referenced by:  2nd1st  8035  cnvf1olem  8105  brtpos  8231  dftpos4  8241  tpostpos  8242  xpcomco  9055  fsumcnv  15824  fprodcnv  16037  gsumcom2  20045  txswaphmeolem  23930  swapf1  49935  swapf2  49937