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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 5585 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
2 | df-br 5058 | . . . 4 ⊢ (𝐴(𝑋 × 𝑋)𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (𝑋 × 𝑋)) | |
3 | 1, 2 | sylibr 235 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → 𝐴(𝑋 × 𝑋)𝐵) |
4 | swoer.1 | . . . . . 6 ⊢ 𝑅 = ((𝑋 × 𝑋) ∖ ( < ∪ ◡ < )) | |
5 | 4 | breqi 5063 | . . . . 5 ⊢ (𝐴𝑅𝐵 ↔ 𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵) |
6 | brdif 5110 | . . . . 5 ⊢ (𝐴((𝑋 × 𝑋) ∖ ( < ∪ ◡ < ))𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) | |
7 | 5, 6 | bitri 276 | . . . 4 ⊢ (𝐴𝑅𝐵 ↔ (𝐴(𝑋 × 𝑋)𝐵 ∧ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
8 | 7 | baib 536 | . . 3 ⊢ (𝐴(𝑋 × 𝑋)𝐵 → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
9 | 3, 8 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ 𝐴( < ∪ ◡ < )𝐵)) |
10 | brun 5108 | . . . 4 ⊢ (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵)) | |
11 | brcnvg 5743 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴◡ < 𝐵 ↔ 𝐵 < 𝐴)) | |
12 | 11 | orbi2d 909 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴 < 𝐵 ∨ 𝐴◡ < 𝐵) ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
13 | 10, 12 | syl5bb 284 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴( < ∪ ◡ < )𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
14 | 13 | notbid 319 | . 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (¬ 𝐴( < ∪ ◡ < )𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
15 | 9, 14 | bitrd 280 | 1 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ↔ ¬ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 841 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ∪ cun 3931 〈cop 4563 class class class wbr 5057 × cxp 5546 ◡ccnv 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-xp 5554 df-cnv 5556 |
This theorem is referenced by: swoer 8308 swoord1 8309 swoord2 8310 swoso 8311 |
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