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Mirrors > Home > MPE Home > Th. List > Mathboxes > atnlt | Structured version Visualization version GIF version |
Description: Two atoms cannot satisfy the less than relation. (Contributed by NM, 7-Feb-2012.) |
Ref | Expression |
---|---|
atnlt.s | ⊢ < = (lt‘𝐾) |
atnlt.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atnlt | ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
2 | 1 | pltirr 18051 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
3 | 2 | 3adant3 1131 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
4 | breq2 5083 | . . . 4 ⊢ (𝑃 = 𝑄 → (𝑃 < 𝑃 ↔ 𝑃 < 𝑄)) | |
5 | 4 | notbid 318 | . . 3 ⊢ (𝑃 = 𝑄 → (¬ 𝑃 < 𝑃 ↔ ¬ 𝑃 < 𝑄)) |
6 | 3, 5 | syl5ibcom 244 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 = 𝑄 → ¬ 𝑃 < 𝑄)) |
7 | eqid 2740 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 18049 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃(le‘𝐾)𝑄)) |
9 | atnlt.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 7, 9 | atcmp 37321 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃(le‘𝐾)𝑄 ↔ 𝑃 = 𝑄)) |
11 | 8, 10 | sylibd 238 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃 = 𝑄)) |
12 | 11 | necon3ad 2958 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ¬ 𝑃 < 𝑄)) |
13 | 6, 12 | pm2.61dne 3033 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1086 = wceq 1542 ∈ wcel 2110 class class class wbr 5079 ‘cfv 6432 lecple 16967 ltcplt 18024 Atomscatm 37273 AtLatcal 37274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-proset 18011 df-poset 18029 df-plt 18046 df-glb 18063 df-p0 18141 df-lat 18148 df-covers 37276 df-ats 37277 df-atl 37308 |
This theorem is referenced by: atltcvr 37445 llnnleat 37523 |
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