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Theorem opswap 6193

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 6187	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩})
2	1	unieqd 4863	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩})
3		opex 5416	. . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V
4	3	unisn 4869	. . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩
5	2, 4	eqtrdi 2787	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩)
6		uni0 4878	. . 3 ⊢ ∪ ∅ = ∅
7		opprc 4839	. . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅)
8	7	sneqd 4579	. . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅})
9	8	cnveqd 5830	. . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ^◡{∅})
10		cnvsn0 6174	. . . . 5 ⊢ ^◡{∅} = ∅
11	9, 10	eqtrdi 2787	. . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ^◡{⟨𝐴, 𝐵⟩} = ∅)
12	11	unieqd 4863	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ^◡{⟨𝐴, 𝐵⟩} = ∪ ∅)
13		ancom 460	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V))
14		opprc 4839	. . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
15	13, 14	sylnbi 330	. . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅)
16	6, 12, 15	3eqtr4a 2797	. 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 1542 ∈ wcel 2114 Vcvv 3429 ∅c0 4273 {csn 4567 ⟨cop 4573 ∪ cuni 4850 ^◡ccnv 5630

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2708 ax-sep 5231 ax-pr 5375

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-xp 5637 df-rel 5638 df-cnv 5639 df-dm 5641 df-rn 5642

This theorem is referenced by: 2nd1st 7991 cnvf1olem 8060 brtpos 8185 dftpos4 8195 tpostpos 8196 xpcomco 9005 fsumcnv 15735 fprodcnv 15948 gsumcom2 19950 txswaphmeolem 23769 swapf1 49747 swapf2 49749