Theorem 4atexlempns 36640

Description: Lemma for 4atexlem7 36653. (Contributed by NM, 23-Nov-2012.)

Hypotheses

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)

Assertion

Ref	Expression
4atexlempns	⊢ (𝜑 → 𝑃 ≠ 𝑆)

Proof of Theorem 4atexlempns

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)

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