Theorem opswap 6230

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 6224	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4880	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5432	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4886	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2812	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4896	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4856	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4596	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5853	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ◡{∅})
10		cnvsn0 6211	. . . . 5 ⊢ ◡{∅} = ∅
11	9, 10	eqtrdi 2812	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4880	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 466	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4856	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 333	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2822	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
17	5, 16	pm2.61i 184	1 ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ cuni 4867  ^◡ccnv 5650

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662

This theorem is used by:  2nd1st  8049  cnvf1olem  8121  brtpos  8252  dftpos4  8262  tpostpos  8263  xpcomco  9086  fsumcnv  15939  fprodcnv  16150  gsumcom2  20189  txswaphmeolem  24123  swapf1  50379  swapf2  50381