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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 6182, 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 6060 | . 2 ⊢ Rel ◡◡{⟨𝐴, 𝐵⟩} | |
2 | relcnv 6060 | . 2 ⊢ Rel ◡{⟨𝐵, 𝐴⟩} | |
3 | vex 3451 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 3451 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 5841 | . . 3 ⊢ (⟨𝑥, 𝑦⟩ ∈ ◡◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩}) |
6 | ancom 462 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | |
7 | 3, 4 | opth 5437 | . . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
8 | 4, 3 | opth 5437 | . . . . . 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 5425 | . . . . . 6 ⊢ ⟨𝑥, 𝑦⟩ ∈ V | |
11 | 10 | elsn 4605 | . . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝐵⟩) |
12 | opex 5425 | . . . . . 6 ⊢ ⟨𝑦, 𝑥⟩ ∈ V | |
13 | 12 | elsn 4605 | . . . . 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 5841 | . . . 4 ⊢ (⟨𝑦, 𝑥⟩ ∈ ◡{⟨𝐴, 𝐵⟩} ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝐴, 𝐵⟩}) |
16 | 3, 4 | opelcnv 5841 | . . . 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 5750 | 1 ⊢ ◡◡{⟨𝐴, 𝐵⟩} = ◡{⟨𝐵, 𝐴⟩} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4590 ⟨cop 4596 ◡ccnv 5636 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 df-cnv 5645 |
This theorem is referenced by: rnsnopg 6177 cnvsng 6179 |
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