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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > llnnlt | Structured version Visualization version GIF version |
Description: Two lattice lines cannot satisfy the less than relation. (Contributed by NM, 26-Jun-2012.) |
Ref | Expression |
---|---|
llnnlt.s | ⊢ < = (lt‘𝐾) |
llnnlt.n | ⊢ 𝑁 = (LLines‘𝐾) |
Ref | Expression |
---|---|
llnnlt | ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | llnnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
2 | 1 | pltirr 17565 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 < 𝑋) |
3 | 2 | 3adant3 1129 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑋) |
4 | breq2 5034 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 < 𝑋 ↔ 𝑋 < 𝑌)) | |
5 | 4 | notbid 321 | . . 3 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 < 𝑋 ↔ ¬ 𝑋 < 𝑌)) |
6 | 3, 5 | syl5ibcom 248 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 = 𝑌 → ¬ 𝑋 < 𝑌)) |
7 | eqid 2798 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 17563 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
9 | llnnlt.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
10 | 7, 9 | llncmp 36818 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋(le‘𝐾)𝑌 ↔ 𝑋 = 𝑌)) |
11 | 8, 10 | sylibd 242 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 < 𝑌 → 𝑋 = 𝑌)) |
12 | 11 | necon3ad 3000 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≠ 𝑌 → ¬ 𝑋 < 𝑌)) |
13 | 6, 12 | pm2.61dne 3073 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 lecple 16564 ltcplt 17543 HLchlt 36646 LLinesclln 36787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 |
This theorem is referenced by: lplnnle2at 36837 |
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