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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 5955 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = {〈𝐵, 𝐴〉}) | |
2 | 1 | unieqd 4755 | . . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ {〈𝐵, 𝐴〉}) |
3 | opex 5248 | . . . 4 ⊢ 〈𝐵, 𝐴〉 ∈ V | |
4 | 3 | unisn 4761 | . . 3 ⊢ ∪ {〈𝐵, 𝐴〉} = 〈𝐵, 𝐴〉 |
5 | 2, 4 | syl6eq 2847 | . 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
6 | uni0 4772 | . . 3 ⊢ ∪ ∅ = ∅ | |
7 | opprc 4733 | . . . . . . 7 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐴, 𝐵〉 = ∅) | |
8 | 7 | sneqd 4484 | . . . . . 6 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → {〈𝐴, 𝐵〉} = {∅}) |
9 | 8 | cnveqd 5632 | . . . . 5 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ◡{∅}) |
10 | cnvsn0 5942 | . . . . 5 ⊢ ◡{∅} = ∅ | |
11 | 9, 10 | syl6eq 2847 | . . . 4 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ◡{〈𝐴, 𝐵〉} = ∅) |
12 | 11 | unieqd 4755 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = ∪ ∅) |
13 | ancom 461 | . . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐵 ∈ V ∧ 𝐴 ∈ V)) | |
14 | opprc 4733 | . . . 4 ⊢ (¬ (𝐵 ∈ V ∧ 𝐴 ∈ V) → 〈𝐵, 𝐴〉 = ∅) | |
15 | 13, 14 | sylnbi 331 | . . 3 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → 〈𝐵, 𝐴〉 = ∅) |
16 | 6, 12, 15 | 3eqtr4a 2857 | . 2 ⊢ (¬ (𝐴 ∈ V ∧ 𝐵 ∈ V) → ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉) |
17 | 5, 16 | pm2.61i 183 | 1 ⊢ ∪ ◡{〈𝐴, 𝐵〉} = 〈𝐵, 𝐴〉 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∅c0 4211 {csn 4472 〈cop 4478 ∪ cuni 4745 ◡ccnv 5442 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pr 5221 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-xp 5449 df-rel 5450 df-cnv 5451 df-dm 5453 df-rn 5454 |
This theorem is referenced by: 2nd1st 7593 cnvf1olem 7661 brtpos 7752 dftpos4 7762 tpostpos 7763 xpcomco 8454 fsumcnv 14961 fprodcnv 15170 gsumcom2 18815 txswaphmeolem 22096 |
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