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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 5539 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 4382 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | opex 4859 | . . . 4 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
4 | 3 | unisn 4387 | . . 3 ⊢ ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
5 | 2, 4 | syl6eq 2660 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
6 | uni0 4401 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | opprc 4362 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
8 | 7 | sneqd 4137 | . . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} = {∅}) |
9 | 8 | cnveqd 5220 | . . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ◡{∅}) |
10 | cnvsn0 5521 | . . . . 5 ⊢ ◡{∅} = ∅ | |
11 | 9, 10 | syl6eq 2660 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ∅) |
12 | 11 | unieqd 4382 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ ∅) |
13 | ancom 465 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
14 | opprc 4362 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 = ∅) | |
15 | 13, 14 | sylnbi 319 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐵, 𝐴〉 = ∅) |
16 | 6, 12, 15 | 3eqtr4a 2670 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
17 | 5, 16 | pm2.61i 175 | 1 ⊢ ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∅c0 3874 {csn 4125 〈cop 4131 ∪ cuni 4372 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 df-rn 5049 |
This theorem is referenced by: 2nd1st 7104 cnvf1olem 7162 brtpos 7248 dftpos4 7258 tpostpos 7259 xpcomco 7935 fsumcnv 14346 fprodcnv 14552 gsumcom2 18197 txswaphmeolem 21417 |
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