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Theorem opswap 6132

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 6126	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4853	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5379	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4861	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2794	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4869	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4827	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4573	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5784	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ^◡{∅})
10		cnvsn0 6113	. . . . 5 ⊢ ^◡{∅} = ∅
11	9, 10	eqtrdi 2794	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4853	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 461	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4827	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 330	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2804	. 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
17	5, 16	pm2.61i 182	1 ⊢ ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∅c0 4256 {csn 4561 ⟨cop 4567 ∪ cuni 4839 ^◡ccnv 5588

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-xp 5595 df-rel 5596 df-cnv 5597 df-dm 5599 df-rn 5600

This theorem is referenced by: 2nd1st 7879 cnvf1olem 7950 brtpos 8051 dftpos4 8061 tpostpos 8062 xpcomco 8849 fsumcnv 15485 fprodcnv 15693 gsumcom2 19576 txswaphmeolem 22955