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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 6069, 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 5952 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 5952 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 3402 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 3402 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 5735 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉}) |
6 | ancom 464 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | |
7 | 3, 4 | opth 5345 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
8 | 4, 3 | opth 5345 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉 ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 6, 7, 8 | 3bitr4i 306 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
10 | opex 5333 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
11 | 10 | elsn 4542 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
12 | opex 5333 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V | |
13 | 12 | elsn 4542 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
14 | 9, 11, 13 | 3bitr4i 306 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
15 | 4, 3 | opelcnv 5735 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 5735 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
17 | 14, 15, 16 | 3bitr4i 306 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
18 | 5, 17 | bitri 278 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
19 | 1, 2, 18 | eqrelriiv 5645 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 {csn 4527 〈cop 4533 ◡ccnv 5535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-rel 5543 df-cnv 5544 |
This theorem is referenced by: rnsnopg 6064 cnvsng 6066 |
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