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Mirrors > Home > ILE Home > Th. List > brdifun | GIF version |
Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.) |
Ref | Expression |
---|---|
swoer.1 | ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) |
Ref | Expression |
---|---|
brdifun | ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opelxpi 4630 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
2 | df-br 3977 | . . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
3 | 1, 2 | sylibr 133 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴(𝑋 × 𝑋)𝐵) |
4 | swoer.1 | . . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) | |
5 | 4 | breqi 3982 | . . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵) |
6 | brdif 4029 | . . . . 5 ⊢ (𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) | |
7 | 5, 6 | bitri 183 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
8 | 7 | baib 909 | . . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
9 | 3, 8 | syl 14 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
10 | brun 4027 | . . . 4 ⊢ (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵)) | |
11 | brcnvg 4779 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)) | |
12 | 11 | orbi2d 780 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
13 | 10, 12 | syl5bb 191 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
14 | 13 | notbid 657 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (¬ 𝐴( < ∪ ◡ < )𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
15 | 9, 14 | bitrd 187 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 = wceq 1342 ∈ wcel 2135 ∖ cdif 3108 ∪ cun 3109 〈cop 3573 class class class wbr 3976 × cxp 4596 ◡ccnv 4597 |
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-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 |
This theorem is referenced by: swoer 6520 swoord1 6521 swoord2 6522 |
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