Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atbtwnexOLDN Structured version   Visualization version   GIF version

Theorem atbtwnexOLDN 37956
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 5126 . . . 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 37895 . . 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 37955 . . . . . . 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 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 ∧ 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 2940  βˆƒwrex 3070   class class class wbr 5106  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  lecple 17145  joincjn 18205  Atomscatm 37771  HLchlt 37858
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-proset 18189  df-poset 18207  df-plt 18224  df-lub 18240  df-glb 18241  df-join 18242  df-meet 18243  df-p0 18319  df-lat 18326  df-clat 18393  df-oposet 37684  df-ol 37686  df-oml 37687  df-covers 37774  df-ats 37775  df-atl 37806  df-cvlat 37830  df-hlat 37859
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator