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Theorem opswap 6227

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 6221	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4916	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5460	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4924	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2783	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4933	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4892	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4636	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5872	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ^◡{∅})
10		cnvsn0 6208	. . . . 5 ⊢ ^◡{∅} = ∅
11	9, 10	eqtrdi 2783	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4916	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 460	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4892	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 330	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2793	. 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 1534 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 {csn 4624 ⟨cop 4630 ∪ cuni 4903 ^◡ccnv 5671

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-dm 5682 df-rn 5683

This theorem is referenced by: 2nd1st 8036 cnvf1olem 8109 brtpos 8234 dftpos4 8244 tpostpos 8245 xpcomco 9080 fsumcnv 15745 fprodcnv 15953 gsumcom2 19923 txswaphmeolem 23701