Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 17573 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁) → ¬ 𝑋 < 𝑋) |
3 | 2 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑋) |
4 | breq2 5070 | . . . 4 ⊢ (𝑋 = 𝑌 → (𝑋 < 𝑋 ↔ 𝑋 < 𝑌)) | |
5 | 4 | notbid 320 | . . 3 ⊢ (𝑋 = 𝑌 → (¬ 𝑋 < 𝑋 ↔ ¬ 𝑋 < 𝑌)) |
6 | 3, 5 | syl5ibcom 247 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 = 𝑌 → ¬ 𝑋 < 𝑌)) |
7 | eqid 2821 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 17571 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
9 | llnnlt.n | . . . . 5 ⊢ 𝑁 = (LLines‘𝐾) | |
10 | 7, 9 | llncmp 36673 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋(le‘𝐾)𝑌 ↔ 𝑋 = 𝑌)) |
11 | 8, 10 | sylibd 241 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 < 𝑌 → 𝑋 = 𝑌)) |
12 | 11 | necon3ad 3029 | . 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → (𝑋 ≠ 𝑌 → ¬ 𝑋 < 𝑌)) |
13 | 6, 12 | pm2.61dne 3103 | 1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) → ¬ 𝑋 < 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5066 ‘cfv 6355 lecple 16572 ltcplt 17551 HLchlt 36501 LLinesclln 36642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-lat 17656 df-clat 17718 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 |
This theorem is referenced by: lplnnle2at 36692 |
Copyright terms: Public domain | W3C validator |