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Mirrors > Home > ILE Home > Th. List > cnvcnvsn | GIF version |
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4991, 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 4887 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 4887 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 2663 | . . . 4 ⊢ 𝑦 ∈ V | |
4 | vex 2663 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | opelcnv 4691 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉}) |
6 | ancom 264 | . . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) | |
7 | 3, 4 | opth 4129 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵)) |
8 | 4, 3 | opth 4129 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉 ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) |
9 | 6, 7, 8 | 3bitr4i 211 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉) |
10 | 3, 4 | opex 4121 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
11 | 10 | elsn 3513 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) |
12 | 4, 3 | opex 4121 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3513 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉) |
14 | 9, 11, 13 | 3bitr4i 211 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉}) |
15 | 4, 3 | opelcnv 4691 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 4691 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉}) |
17 | 14, 15, 16 | 3bitr4i 211 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉}) |
18 | 5, 17 | bitri 183 | . 2 ⊢ (〈𝑦, 𝑥〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉}) |
19 | 1, 2, 18 | eqrelriiv 4603 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1316 ∈ wcel 1465 {csn 3497 〈cop 3500 ◡ccnv 4508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-br 3900 df-opab 3960 df-xp 4515 df-rel 4516 df-cnv 4517 |
This theorem is referenced by: rnsnopg 4987 cnvsn 4991 |
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