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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 5618, 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 5503 | . 2 ⊢ Rel ◡◡{〈𝐴, 𝐵〉} | |
2 | relcnv 5503 | . 2 ⊢ Rel ◡{〈𝐵, 𝐴〉} | |
3 | vex 3203 | . . . 4 ⊢ 𝑥 ∈ V | |
4 | vex 3203 | . . . 4 ⊢ 𝑦 ∈ V | |
5 | 3, 4 | opelcnv 5304 | . . 3 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉}) |
6 | ancom 466 | . . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) | |
7 | 3, 4 | opth 4945 | . . . . . 6 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) |
8 | 4, 3 | opth 4945 | . . . . . 6 ⊢ (〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉 ↔ (𝑦 = 𝐵 ∧ 𝑥 = 𝐴)) |
9 | 6, 7, 8 | 3bitr4i 292 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉 ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
10 | opex 4932 | . . . . . 6 ⊢ 〈𝑥, 𝑦〉 ∈ V | |
11 | 10 | elsn 4192 | . . . . 5 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 = 〈𝐴, 𝐵〉) |
12 | opex 4932 | . . . . . 6 ⊢ 〈𝑦, 𝑥〉 ∈ V | |
13 | 12 | elsn 4192 | . . . . 5 ⊢ (〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 = 〈𝐵, 𝐴〉) |
14 | 9, 11, 13 | 3bitr4i 292 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
15 | 4, 3 | opelcnv 5304 | . . . 4 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ {〈𝐴, 𝐵〉}) |
16 | 3, 4 | opelcnv 5304 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉} ↔ 〈𝑦, 𝑥〉 ∈ {〈𝐵, 𝐴〉}) |
17 | 14, 15, 16 | 3bitr4i 292 | . . 3 ⊢ (〈𝑦, 𝑥〉 ∈ ◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
18 | 5, 17 | bitri 264 | . 2 ⊢ (〈𝑥, 𝑦〉 ∈ ◡◡{〈𝐴, 𝐵〉} ↔ 〈𝑥, 𝑦〉 ∈ ◡{〈𝐵, 𝐴〉}) |
19 | 1, 2, 18 | eqrelriiv 5214 | 1 ⊢ ◡◡{〈𝐴, 𝐵〉} = ◡{〈𝐵, 𝐴〉} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 {csn 4177 〈cop 4183 ◡ccnv 5113 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-cnv 5122 |
This theorem is referenced by: rnsnopg 5614 cnvsn 5618 strlemor1OLD 15969 |
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