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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 6245 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 4925 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | opex 5475 | . . . 4 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
4 | 3 | unisn 4931 | . . 3 ⊢ ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
5 | 2, 4 | eqtrdi 2791 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
6 | uni0 4940 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | opprc 4901 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
8 | 7 | sneqd 4643 | . . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} = {∅}) |
9 | 8 | cnveqd 5889 | . . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ◡{∅}) |
10 | cnvsn0 6232 | . . . . 5 ⊢ ◡{∅} = ∅ | |
11 | 9, 10 | eqtrdi 2791 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ∅) |
12 | 11 | unieqd 4925 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ ∅) |
13 | ancom 460 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
14 | opprc 4901 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 = ∅) | |
15 | 13, 14 | sylnbi 330 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐵, 𝐴〉 = ∅) |
16 | 6, 12, 15 | 3eqtr4a 2801 | . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 〈cop 4637 ∪ cuni 4912 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 df-rn 5700 |
This theorem is referenced by: 2nd1st 8062 cnvf1olem 8134 brtpos 8259 dftpos4 8269 tpostpos 8270 xpcomco 9101 fsumcnv 15806 fprodcnv 16016 gsumcom2 20008 txswaphmeolem 23828 |
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