opswap - Metamath Proof Explorer

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Theorem opswap 6218

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 6212	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4896	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5439	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4902	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2786	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4911	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4872	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4613	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5855	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ^◡{∅})
10		cnvsn0 6199	. . . . 5 ⊢ ^◡{∅} = ∅
11	9, 10	eqtrdi 2786	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4896	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 460	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4872	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 330	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2796	. 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 395 = wceq 1540 ∈ wcel 2108 Vcvv 3459 ∅c0 4308 {csn 4601 ⟨cop 4607 ∪ cuni 4883 ^◡ccnv 5653

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-xp 5660 df-rel 5661 df-cnv 5662 df-dm 5664 df-rn 5665

This theorem is referenced by: 2nd1st 8037 cnvf1olem 8109 brtpos 8234 dftpos4 8244 tpostpos 8245 xpcomco 9076 fsumcnv 15789 fprodcnv 15999 gsumcom2 19956 txswaphmeolem 23742 swapf1 49189 swapf2 49191