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Theorem atbtwnexOLDN 38318
Description: There exists a 3rd atom π‘Ÿ on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃, such that π‘Ÿ is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
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
atbtwnexOLDN (((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 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
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 atbtwnexOLDN
StepHypRef Expression
1 simpr2 1196 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ 𝑃 ≀ 𝑋)
2 simpr3 1197 . . . 4 (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ 𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋)) β†’ Β¬ 𝑄 ≀ 𝑋)
3 nbrne2 5169 . . . 4 ((𝑃 ≀ 𝑋 ∧ Β¬ 𝑄 ≀ 𝑋) β†’ 𝑃 β‰  𝑄)
41, 2, 3syl2anc 585 . . 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 38257 . . 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 592 . 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 1211 . . . . 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 1210 . . . . . 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 1198 . . . . . . 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 1138 . . . . . . 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 1270 . . . . . . 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 1271 . . . . . . 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 1272 . . . . . . 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 1212 . . . . . . 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 38317 . . . . . . 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 1388 . . . . . 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)
2210, 21, 173jca 1129 . . . 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 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
23223exp 1120 . . 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 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))))
2423reximdvai 3166 . 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 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄))))
259, 24mpd 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 ∧ π‘Ÿ ≀ (𝑃 ∨ 𝑄)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆƒwrex 3071   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  lecple 17204  joincjn 18264  Atomscatm 38133  HLchlt 38220
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221
This theorem is referenced by: (None)
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