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Theorem atbtwnex 38832
Description: Given atoms 𝑃 in 𝑋 and 𝑄 not in 𝑋, there exists an atom π‘Ÿ not in 𝑋 such that the line 𝑄 ∨ π‘Ÿ intersects 𝑋 at 𝑃. (Contributed by NM, 1-Aug-2012.)
Hypotheses
Ref Expression
atbtwn.b 𝐡 = (Baseβ€˜πΎ)
atbtwn.l ≀ = (leβ€˜πΎ)
atbtwn.j ∨ = (joinβ€˜πΎ)
atbtwn.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
atbtwnex (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
Distinct variable groups:   𝐴,π‘Ÿ   𝐡,π‘Ÿ   𝐾,π‘Ÿ   ≀ ,π‘Ÿ   𝑃,π‘Ÿ   𝑄,π‘Ÿ   𝑋,π‘Ÿ
Allowed substitution hint:   ∨ (π‘Ÿ)

Proof of Theorem atbtwnex
StepHypRef Expression
1 simpr2 1192 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ 𝑋)
2 simpr3 1193 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑄 ≀ 𝑋)
3 nbrne2 5161 . . . 4 ((𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑃 β‰  𝑄)
41, 2, 3syl2anc 583 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 β‰  𝑄)
5 atbtwn.l . . . 4 ≀ = (leβ€˜πΎ)
6 atbtwn.j . . . 4 ∨ = (joinβ€˜πΎ)
7 atbtwn.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
85, 6, 7hlsupr 38770 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
94, 8syldan 590 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
10 simp32 1207 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ π‘Ÿ β‰  𝑄)
11 simp31 1206 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ π‘Ÿ β‰  𝑃)
12 simp1l 1194 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴))
13 simp2 1134 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ π‘Ÿ ∈ 𝐴)
14 simp1r1 1266 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑋 ∈ 𝐡)
15 simp1r2 1267 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ≀ 𝑋)
16 simp1r3 1268 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑄 ≀ 𝑋)
17 simp33 1208 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ π‘Ÿ ≀ (𝑃 ∨ 𝑄))
18 atbtwn.b . . . . . . . 8 𝐡 = (Baseβ€˜πΎ)
1918, 5, 6, 7atbtwn 38830 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑋 ∈ 𝐡) ∧ (𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (π‘Ÿ β‰  𝑃 ↔ Β¬ π‘Ÿ ≀ 𝑋))
2012, 13, 14, 15, 16, 17, 19syl123anc 1384 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (π‘Ÿ β‰  𝑃 ↔ Β¬ π‘Ÿ ≀ 𝑋))
2111, 20mpbid 231 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ π‘Ÿ ≀ 𝑋)
22 simp1l1 1263 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝐾 ∈ HL)
23 simp1l2 1264 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ∈ 𝐴)
24 simp1l3 1265 . . . . . . . 8 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑄 ∈ 𝐴)
255, 6, 7hlatexch2 38780 . . . . . . . 8 ((𝐾 ∈ HL ∧ (π‘Ÿ ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ π‘Ÿ β‰  𝑄) β†’ (π‘Ÿ ≀ (𝑃 ∨ 𝑄) β†’ 𝑃 ≀ (π‘Ÿ ∨ 𝑄)))
2622, 13, 23, 24, 10, 25syl131anc 1380 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (π‘Ÿ ≀ (𝑃 ∨ 𝑄) β†’ 𝑃 ≀ (π‘Ÿ ∨ 𝑄)))
2717, 26mpd 15 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ≀ (π‘Ÿ ∨ 𝑄))
286, 7hlatjcom 38751 . . . . . . 7 ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴) β†’ (𝑄 ∨ π‘Ÿ) = (π‘Ÿ ∨ 𝑄))
2922, 24, 13, 28syl3anc 1368 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (𝑄 ∨ π‘Ÿ) = (π‘Ÿ ∨ 𝑄))
3027, 29breqtrrd 5169 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))
3110, 21, 303jca 1125 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) ∧ π‘Ÿ ∈ 𝐴 ∧ (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))) β†’ (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
32313exp 1116 . . 3 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (π‘Ÿ ∈ 𝐴 β†’ ((π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)) β†’ (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))))
3332reximdvai 3159 . 2 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ (βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑃 ∧ π‘Ÿ β‰  𝑄 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ))))
349, 33mpd 15 1 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ βˆƒπ‘Ÿ ∈ 𝐴 (π‘Ÿ β‰  𝑄 ∧ Β¬ π‘Ÿ ≀ 𝑋 ∧ 𝑃 ≀ (𝑄 ∨ π‘Ÿ)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   class class class wbr 5141  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  lecple 17213  joincjn 18276  Atomscatm 38646  HLchlt 38733
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-proset 18260  df-poset 18278  df-plt 18295  df-lub 18311  df-glb 18312  df-join 18313  df-meet 18314  df-p0 18390  df-lat 18397  df-clat 18464  df-oposet 38559  df-ol 38561  df-oml 38562  df-covers 38649  df-ats 38650  df-atl 38681  df-cvlat 38705  df-hlat 38734
This theorem is referenced by:  dalem19  39066
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