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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 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 |
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