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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 17576 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
3 | 2 | 3adant3 1128 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑃) |
4 | breq2 5073 | . . . 4 ⊢ (𝑃 = 𝑄 → (𝑃 < 𝑃 ↔ 𝑃 < 𝑄)) | |
5 | 4 | notbid 320 | . . 3 ⊢ (𝑃 = 𝑄 → (¬ 𝑃 < 𝑃 ↔ ¬ 𝑃 < 𝑄)) |
6 | 3, 5 | syl5ibcom 247 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 = 𝑄 → ¬ 𝑃 < 𝑄)) |
7 | eqid 2824 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
8 | 7, 1 | pltle 17574 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃(le‘𝐾)𝑄)) |
9 | atnlt.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 7, 9 | atcmp 36451 | . . . 4 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃(le‘𝐾)𝑄 ↔ 𝑃 = 𝑄)) |
11 | 8, 10 | sylibd 241 | . . 3 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 < 𝑄 → 𝑃 = 𝑄)) |
12 | 11 | necon3ad 3032 | . 2 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ≠ 𝑄 → ¬ 𝑃 < 𝑄)) |
13 | 6, 12 | pm2.61dne 3106 | 1 ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → ¬ 𝑃 < 𝑄) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 lecple 16575 ltcplt 17554 Atomscatm 36403 AtLatcal 36404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-proset 17541 df-poset 17559 df-plt 17571 df-glb 17588 df-p0 17652 df-lat 17659 df-covers 36406 df-ats 36407 df-atl 36438 |
This theorem is referenced by: atltcvr 36575 llnnleat 36653 |
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