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Mirrors > Home > MPE Home > Th. List > cnvcnvsn | Structured version Visualization version GIF version |
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 5922, this does not need any sethood assumptions on 𝐴 and 𝐵.) (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
cnvcnvsn | ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 5807 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 5807 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 3418 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 3418 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 5602 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉}) |
6 | ancom 453 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | |
7 | 3, 4 | opth 5225 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
8 | 4, 3 | opth 5225 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉 ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 6, 7, 8 | 3bitr4i 295 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
10 | opex 5213 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
11 | 10 | elsn 4456 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
12 | opex 5213 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V | |
13 | 12 | elsn 4456 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
14 | 9, 11, 13 | 3bitr4i 295 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
15 | 4, 3 | opelcnv 5602 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 5602 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
17 | 14, 15, 16 | 3bitr4i 295 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
18 | 5, 17 | bitri 267 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
19 | 1, 2, 18 | eqrelriiv 5513 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 387 = wceq 1507 ∈ wcel 2050 {csn 4441 〈cop 4447 ◡ccnv 5406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-rab 3097 df-v 3417 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-br 4930 df-opab 4992 df-xp 5413 df-rel 5414 df-cnv 5415 |
This theorem is referenced by: rnsnopg 5917 cnvsng 5919 |
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