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Mirrors > Home > MPE Home > Th. List > opswap | Structured version Visualization version GIF version |
Description: Swap the members of an ordered pair. (Contributed by NM, 14-Dec-2008.) (Revised by Mario Carneiro, 30-Aug-2015.) |
Ref | Expression |
---|---|
opswap | ⊢ ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvsng 6222 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = {⟨𝐵, 𝐴⟩}) | |
2 | 1 | unieqd 4922 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ {⟨𝐵, 𝐴⟩}) |
3 | opex 5464 | . . . 4 ⊢ ⟨𝐵, 𝐴⟩ ∈ V | |
4 | 3 | unisn 4930 | . . 3 ⊢ ∪ {⟨𝐵, 𝐴⟩} = ⟨𝐵, 𝐴⟩ |
5 | 2, 4 | eqtrdi 2788 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ⟨𝐵, 𝐴⟩) |
6 | uni0 4939 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | opprc 4896 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐴, 𝐵⟩ = ∅) | |
8 | 7 | sneqd 4640 | . . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = {∅}) |
9 | 8 | cnveqd 5875 | . . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ◡{∅}) |
10 | cnvsn0 6209 | . . . . 5 ⊢ ◡{∅} = ∅ | |
11 | 9, 10 | eqtrdi 2788 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{⟨𝐴, 𝐵⟩} = ∅) |
12 | 11 | unieqd 4922 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{⟨𝐴, 𝐵⟩} = ∪ ∅) |
13 | ancom 461 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
14 | opprc 4896 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅) | |
15 | 13, 14 | sylnbi 329 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ⟨𝐵, 𝐴⟩ = ∅) |
16 | 6, 12, 15 | 3eqtr4a 2798 | . 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 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 {csn 4628 ⟨cop 4634 ∪ cuni 4908 ◡ccnv 5675 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 |
This theorem is referenced by: 2nd1st 8023 cnvf1olem 8095 brtpos 8219 dftpos4 8229 tpostpos 8230 xpcomco 9061 fsumcnv 15718 fprodcnv 15926 gsumcom2 19842 txswaphmeolem 23307 |
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