Metamath Proof Explorer

Theorem cdleme40m

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT Use proof idea from cdleme32sn1awN . (Contributed by NM, 18-Mar-2013)

Ref Expression
Hypotheses cdleme40.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme40.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme40.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme40.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme40.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme40.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme40.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme40.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40.i ⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) )
cdleme40.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 )
cdleme40a1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40a1.c ⊢ 𝐶 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) )
cdleme40.t ⊢ 𝑇 = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdleme40.f ⊢ 𝐹 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑆 ∨ 𝑣 ) ∧ 𝑊 ) ) )
Assertion cdleme40m ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 )

Proof

Step Hyp Ref Expression
1 cdleme40.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme40.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme40.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme40.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme40.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme40.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme40.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme40.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdleme40.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdleme40.i ⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) )
11 cdleme40.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 )
12 cdleme40a1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
13 cdleme40a1.c ⊢ 𝐶 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) )
14 cdleme40.t ⊢ 𝑇 = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) )
15 cdleme40.f ⊢ 𝐹 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑆 ∨ 𝑣 ) ∧ 𝑊 ) ) )
16 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ∈ 𝐴 )
17 simp3l1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
18 9 10 11 12 13 cdleme31sn1c ⊢ ( ( 𝑅 ∈ 𝐴 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = 𝐶 )
19 16 17 18 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 = 𝐶 )
20 1 fvexi ⊢ 𝐵 ∈ V
21 nfv ⊢ Ⅎ 𝑡 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
22 nfra1 ⊢ Ⅎ 𝑡 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 )
23 nfcv ⊢ Ⅎ 𝑡 𝐵
24 22 23 nfriota ⊢ Ⅎ 𝑡 ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) )
25 13 24 nfcxfr ⊢ Ⅎ 𝑡 𝐶
26 nfcv ⊢ Ⅎ 𝑡 𝐹
27 25 26 nfne ⊢ Ⅎ 𝑡 𝐶 ≠ 𝐹
28 27 a1i ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → Ⅎ 𝑡 𝐶 ≠ 𝐹 )
29 13 a1i ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐶 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) ) )
30 neeq1 ⊢ ( 𝑌 = 𝐶 → ( 𝑌 ≠ 𝐹 ↔ 𝐶 ≠ 𝐹 ) )
31 30 adantl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ 𝑌 = 𝐶 ) → ( 𝑌 ≠ 𝐹 ↔ 𝐶 ≠ 𝐹 ) )
32 simpl1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
33 simpl2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) )
34 simpl3l ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) )
35 simprl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑡 ∈ 𝐴 )
36 simprrl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑡 ≤ 𝑊 )
37 simprrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
38 35 36 37 jca31 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
39 simp3r1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑣 ∈ 𝐴 )
40 simp3r2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑣 ≤ 𝑊 )
41 simp3r3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) )
42 39 40 41 jca31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ) ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) )
43 42 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ) ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) )
44 2 3 4 5 6 7 8 12 14 15 cdleme39n ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ) ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑌 ≠ 𝐹 )
45 32 33 34 38 43 44 syl113anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑌 ≠ 𝐹 )
46 45 ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑌 ≠ 𝐹 ) )
47 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
48 simp22r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑅 ≤ 𝑊 )
49 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
50 1 2 3 4 5 6 7 8 12 13 cdleme25cl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐶 ∈ 𝐵 )
51 47 16 48 49 17 50 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐶 ∈ 𝐵 )
52 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
53 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
54 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
55 2 3 5 6 cdlemb2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑡 ∈ 𝐴 ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
56 52 53 54 49 55 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ∃ 𝑡 ∈ 𝐴 ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
57 21 28 29 31 46 51 56 riotasv3d ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) ∧ 𝐵 ∈ V ) → 𝐶 ≠ 𝐹 )
58 20 57 mpan2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐶 ≠ 𝐹 )
59 19 58 eqnetrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 )