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Mirrors > Home > MPE Home > Th. List > brdifun | Structured version Visualization version 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 5072 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
2 | df-br 4584 | . . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
3 | 1, 2 | sylibr 223 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴(𝑋 × 𝑋)𝐵) |
4 | swoer.1 | . . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) | |
5 | 4 | breqi 4589 | . . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵) |
6 | brdif 4635 | . . . . 5 ⊢ (𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) | |
7 | 5, 6 | bitri 263 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
8 | 7 | baib 942 | . . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
9 | 3, 8 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
10 | brun 4633 | . . . 4 ⊢ (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵)) | |
11 | brcnvg 5225 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)) | |
12 | 11 | orbi2d 734 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
13 | 10, 12 | syl5bb 271 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
14 | 13 | notbid 307 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (¬ 𝐴( < ∪ ◡ < )𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
15 | 9, 14 | bitrd 267 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∪ cun 3538 〈cop 4131 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 |
This theorem is referenced by: swoer 7659 swoord1 7660 swoord2 7661 swoso 7662 |
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