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Theorem atbtwnexOLDN 36452
 Description: There exists a 3rd atom 𝑟 on a line 𝑃 ∨ 𝑄 intersecting element 𝑋 at 𝑃, such that 𝑟 is different from 𝑄 and not in 𝑋. (Contributed by NM, 30-Jul-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
atbtwn.b 𝐵 = (Base‘𝐾)
atbtwn.l = (le‘𝐾)
atbtwn.j = (join‘𝐾)
atbtwn.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atbtwnexOLDN (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴 (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄)))
Distinct variable groups:   𝐴,𝑟   𝐵,𝑟   𝐾,𝑟   ,𝑟   𝑃,𝑟   𝑄,𝑟   𝑋,𝑟
Allowed substitution hint:   (𝑟)

Proof of Theorem atbtwnexOLDN
StepHypRef Expression
1 simpr2 1189 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃 𝑋)
2 simpr3 1190 . . . 4 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ¬ 𝑄 𝑋)
3 nbrne2 5082 . . . 4 ((𝑃 𝑋 ∧ ¬ 𝑄 𝑋) → 𝑃𝑄)
41, 2, 3syl2anc 584 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → 𝑃𝑄)
5 atbtwn.l . . . 4 = (le‘𝐾)
6 atbtwn.j . . . 4 = (join‘𝐾)
7 atbtwn.a . . . 4 𝐴 = (Atoms‘𝐾)
85, 6, 7hlsupr 36391 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
94, 8syldan 591 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)))
10 simp32 1204 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑟𝑄)
11 simp31 1203 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑟𝑃)
12 simp1l 1191 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → (𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴))
13 simp2 1131 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑟𝐴)
14 simp1r1 1263 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑋𝐵)
15 simp1r2 1264 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑃 𝑋)
16 simp1r3 1265 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → ¬ 𝑄 𝑋)
17 simp33 1205 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → 𝑟 (𝑃 𝑄))
18 atbtwn.b . . . . . . . 8 𝐵 = (Base‘𝐾)
1918, 5, 6, 7atbtwn 36451 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑟𝐴𝑋𝐵) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋𝑟 (𝑃 𝑄))) → (𝑟𝑃 ↔ ¬ 𝑟 𝑋))
2012, 13, 14, 15, 16, 17, 19syl123anc 1381 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → (𝑟𝑃 ↔ ¬ 𝑟 𝑋))
2111, 20mpbid 233 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → ¬ 𝑟 𝑋)
2210, 21, 173jca 1122 . . . 4 ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ 𝑟𝐴 ∧ (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄))) → (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄)))
23223exp 1113 . . 3 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (𝑟𝐴 → ((𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)) → (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄)))))
2423reximdvai 3276 . 2 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → (∃𝑟𝐴 (𝑟𝑃𝑟𝑄𝑟 (𝑃 𝑄)) → ∃𝑟𝐴 (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄))))
259, 24mpd 15 1 (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴 (𝑟𝑄 ∧ ¬ 𝑟 𝑋𝑟 (𝑃 𝑄)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107   ≠ wne 3020  ∃wrex 3143   class class class wbr 5062  ‘cfv 6351  (class class class)co 7151  Basecbs 16475  lecple 16564  joincjn 17546  Atomscatm 36268  HLchlt 36355 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-lat 17648  df-clat 17710  df-oposet 36181  df-ol 36183  df-oml 36184  df-covers 36271  df-ats 36272  df-atl 36303  df-cvlat 36327  df-hlat 36356 This theorem is referenced by: (None)
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