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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 6079 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 4851 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | opex 5355 | . . . 4 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
4 | 3 | unisn 4857 | . . 3 ⊢ ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
5 | 2, 4 | syl6eq 2872 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
6 | uni0 4865 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | opprc 4825 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
8 | 7 | sneqd 4578 | . . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} = {∅}) |
9 | 8 | cnveqd 5745 | . . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ◡{∅}) |
10 | cnvsn0 6066 | . . . . 5 ⊢ ◡{∅} = ∅ | |
11 | 9, 10 | syl6eq 2872 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ∅) |
12 | 11 | unieqd 4851 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ ∅) |
13 | ancom 463 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
14 | opprc 4825 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 = ∅) | |
15 | 13, 14 | sylnbi 332 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐵, 𝐴〉 = ∅) |
16 | 6, 12, 15 | 3eqtr4a 2882 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
17 | 5, 16 | pm2.61i 184 | 1 ⊢ ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 {csn 4566 〈cop 4572 ∪ cuni 4837 ◡ccnv 5553 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-xp 5560 df-rel 5561 df-cnv 5562 df-dm 5564 df-rn 5565 |
This theorem is referenced by: 2nd1st 7736 cnvf1olem 7804 brtpos 7900 dftpos4 7910 tpostpos 7911 xpcomco 8606 fsumcnv 15127 fprodcnv 15336 gsumcom2 19094 txswaphmeolem 22411 |
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