Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > cnvcnvsn | GIF version |
Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 4823, 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 4723 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 4723 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 2604 | . . . 4 ⊢ 𝑦 ∈ V | |
4 | vex 2604 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | opelcnv 4535 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉}) |
6 | ancom 262 | . . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 = 𝐵) ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) | |
7 | 3, 4 | opth 3992 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵)) |
8 | 4, 3 | opth 3992 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉 ↔ (𝑥 = 𝐵 ∧ 𝑦 = 𝐴)) |
9 | 6, 7, 8 | 3bitr4i 210 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉) |
10 | 3, 4 | opex 3984 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
11 | 10 | elsn 3414 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) |
12 | 4, 3 | opex 3984 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3414 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐵, 𝐴〉) |
14 | 9, 11, 13 | 3bitr4i 210 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉}) |
15 | 4, 3 | opelcnv 4535 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 4535 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐵, 𝐴〉}) |
17 | 14, 15, 16 | 3bitr4i 210 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉}) |
18 | 5, 17 | bitri 182 | . 2 ⊢ (〈𝑦, 𝑥〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐵, 𝐴〉}) |
19 | 1, 2, 18 | eqrelriiv 4452 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∈ wcel 1433 {csn 3398 〈cop 3401 ◡ccnv 4362 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-br 3786 df-opab 3840 df-xp 4369 df-rel 4370 df-cnv 4371 |
This theorem is referenced by: rnsnopg 4819 cnvsn 4823 |
Copyright terms: Public domain | W3C validator |