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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 5093, 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 4989 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 4989 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 2733 | . . . 4 ⊢ 𝑦 ∈ V | |
4 | vex 2733 | . . . 4 ⊢ 𝑥 ∈ V | |
5 | 3, 4 | opelcnv 4793 | . . 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 4222 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉 ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐵)) |
8 | 4, 3 | opth 4222 | . . . . . 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 4214 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V |
11 | 10 | elsn 3599 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐴, 𝐵〉) |
12 | 4, 3 | opex 4214 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V |
13 | 12 | elsn 3599 | . . . . 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 4793 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 4793 | . . . 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 4705 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1348 ∈ wcel 2141 {csn 3583 〈cop 3586 ◡ccnv 4610 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-br 3990 df-opab 4051 df-xp 4617 df-rel 4618 df-cnv 4619 |
This theorem is referenced by: rnsnopg 5089 cnvsn 5093 |
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