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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 16786 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
3 | 2 | 3adant3 1074 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
4 | breq2 4587 | . . . 4 ⊢ (𝑃 = 𝑄 → (𝑃 < 𝑃 ↔ 𝑃 < 𝑄)) | |
5 | 4 | notbid 307 | . . 3 ⊢ (𝑃 = 𝑄 → (¬ 𝑃 < 𝑃 ↔ ¬ 𝑃 < 𝑄)) |
6 | 3, 5 | syl5ibcom 234 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 = 𝑄 → ¬ 𝑃 < 𝑄)) |
7 | eqid 2610 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 16784 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃(le‘𝐾)𝑄)) |
9 | atnlt.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 7, 9 | atcmp 33616 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃(le‘𝐾)𝑄 ↔ 𝑃 = 𝑄)) |
11 | 8, 10 | sylibd 228 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃 = 𝑄)) |
12 | 11 | necon3ad 2795 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ¬ 𝑃 < 𝑄)) |
13 | 6, 12 | pm2.61dne 2868 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 lecple 15775 ltcplt 16764 Atomscatm 33568 AtLatcal 33569 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-preset 16751 df-poset 16769 df-plt 16781 df-glb 16798 df-p0 16862 df-lat 16869 df-covers 33571 df-ats 33572 df-atl 33603 |
This theorem is referenced by: atltcvr 33739 llnnleat 33817 |
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