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| Description: Double converse of a singleton of an ordered pair. (Unlike cnvsn 6246, 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 6122 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
| 2 | relcnv 6122 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
| 3 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 4 | vex 3484 | . . . 4 ⊢ 𝑦 ∈ V | |
| 5 | 3, 4 | opelcnv 5892 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉}) | 
| 6 | ancom 460 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | |
| 7 | 3, 4 | opth 5481 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) | 
| 8 | 4, 3 | opth 5481 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉 ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | 
| 9 | 6, 7, 8 | 3bitr4i 303 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) | 
| 10 | opex 5469 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
| 11 | 10 | elsn 4641 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) | 
| 12 | opex 5469 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V | |
| 13 | 12 | elsn 4641 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) | 
| 14 | 9, 11, 13 | 3bitr4i 303 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) | 
| 15 | 4, 3 | opelcnv 5892 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) | 
| 16 | 3, 4 | opelcnv 5892 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) | 
| 17 | 14, 15, 16 | 3bitr4i 303 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) | 
| 18 | 5, 17 | bitri 275 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) | 
| 19 | 1, 2, 18 | eqrelriiv 5800 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2108 {csn 4626 〈cop 4632 ◡ccnv 5684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 | 
| This theorem is referenced by: rnsnopg 6241 cnvsng 6243 | 
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