![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > 4atexlempns | Structured version Visualization version GIF version |
Description: Lemma for 4atexlem7 36653. (Contributed by NM, 23-Nov-2012.) |
Ref | Expression |
---|---|
4thatlem.ph | ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) |
4thatlemslps.l | ⊢ ≤ = (le‘𝐾) |
4thatlemslps.j | ⊢ ∨ = (join‘𝐾) |
4thatlemslps.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
4atexlempns | ⊢ (𝜑 → 𝑃 ≠ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4thatlem.ph | . . 3 ⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑆 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ (𝑃 ∨ 𝑅) = (𝑄 ∨ 𝑅)) ∧ (𝑇 ∈ 𝐴 ∧ (𝑈 ∨ 𝑇) = (𝑉 ∨ 𝑇))) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)))) | |
2 | 1 | 4atexlemk 36625 | . 2 ⊢ (𝜑 → 𝐾 ∈ HL) |
3 | 1 | 4atexlemp 36628 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
4 | 1 | 4atexlemq 36629 | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
5 | 1 | 4atexlems 36630 | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝐴) |
6 | 1 | 4atexlemnslpq 36634 | . 2 ⊢ (𝜑 → ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) |
7 | 4thatlemslps.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
8 | 4thatlemslps.j | . . 3 ⊢ ∨ = (join‘𝐾) | |
9 | 4thatlemslps.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | 7, 8, 9 | 4atlem0be 36173 | . 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑆) |
11 | 2, 3, 4, 5, 6, 10 | syl131anc 1363 | 1 ⊢ (𝜑 → 𝑃 ≠ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ≠ wne 2968 class class class wbr 4929 ‘cfv 6188 (class class class)co 6976 lecple 16428 joincjn 17412 Atomscatm 35841 HLchlt 35928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-lub 17442 df-join 17444 df-lat 17514 df-ats 35845 df-atl 35876 df-cvlat 35900 df-hlat 35929 |
This theorem is referenced by: 4atexlemv 36643 4atexlemc 36647 4atexlemnclw 36648 4atexlemex2 36649 |
Copyright terms: Public domain | W3C validator |