Theorem 4atexlempns 38003

Description: Lemma for 4atexlem7 38016. (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 37988	. 2 ⊢ (𝜑 → 𝐾 ∈ HL)
3	1	4atexlemp 37991	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
4	1	4atexlemq 37992	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
5	1	4atexlems 37993	. 2 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
6	1	4atexlemnslpq 37997	. 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 37536	. 2 ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴) ∧ ¬ 𝑆 ≤ (𝑃 ∨ 𝑄)) → 𝑃 ≠ 𝑆)
11	2, 3, 4, 5, 6, 10	syl131anc 1381	1 ⊢ (𝜑 → 𝑃 ≠ 𝑆)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942   class class class wbr 5070  ‘cfv 6418  (class class class)co 7255  lecple 16895  joincjn 17944  Atomscatm 37204  HLchlt 37291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-lub 17979  df-join 17981  df-lat 18065  df-ats 37208  df-atl 37239  df-cvlat 37263  df-hlat 37292

This theorem is referenced by:  4atexlemv  38006  4atexlemc  38010  4atexlemnclw  38011  4atexlemex2  38012