opswap - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > opswap		Structured version Visualization version GIF version

Theorem opswap 6249

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 6243	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4920	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5469	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4926	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2793	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4935	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4896	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4638	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5886	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ^◡{∅})
10		cnvsn0 6230	. . . . 5 ⊢ ^◡{∅} = ∅
11	9, 10	eqtrdi 2793	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4920	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 460	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4896	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 330	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2803	. 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 3480 ∅c0 4333 {csn 4626 ⟨cop 4632 ∪ cuni 4907 ^◡ccnv 5684

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 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696

This theorem is referenced by: 2nd1st 8063 cnvf1olem 8135 brtpos 8260 dftpos4 8270 tpostpos 8271 xpcomco 9102 fsumcnv 15809 fprodcnv 16019 gsumcom2 19993 txswaphmeolem 23812 swapf1 48978 swapf2 48980