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Theorem oppne2 28426
Description: Points lying on opposite sides of a line cannot be on the line. (Contributed by Thierry Arnoux, 3-Mar-2020.)
Hypotheses
Ref Expression
hpg.p 𝑃 = (Baseβ€˜πΊ)
hpg.d βˆ’ = (distβ€˜πΊ)
hpg.i 𝐼 = (Itvβ€˜πΊ)
hpg.o 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
opphl.l 𝐿 = (LineGβ€˜πΊ)
opphl.d (πœ‘ β†’ 𝐷 ∈ ran 𝐿)
opphl.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
oppcom.a (πœ‘ β†’ 𝐴 ∈ 𝑃)
oppcom.b (πœ‘ β†’ 𝐡 ∈ 𝑃)
oppcom.o (πœ‘ β†’ 𝐴𝑂𝐡)
Assertion
Ref Expression
oppne2 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
Distinct variable groups:   𝐷,π‘Ž,𝑏   𝐼,π‘Ž,𝑏   𝑃,π‘Ž,𝑏   𝑑,𝐴   𝑑,𝐡   𝑑,𝐷   𝑑,𝐺   𝑑,𝐿   𝑑,𝐼   𝑑,𝑂   𝑑,𝑃   πœ‘,𝑑   𝑑, βˆ’   𝑑,π‘Ž,𝑏
Allowed substitution hints:   πœ‘(π‘Ž,𝑏)   𝐴(π‘Ž,𝑏)   𝐡(π‘Ž,𝑏)   𝐺(π‘Ž,𝑏)   𝐿(π‘Ž,𝑏)   βˆ’ (π‘Ž,𝑏)   𝑂(π‘Ž,𝑏)

Proof of Theorem oppne2
StepHypRef Expression
1 oppcom.o . . 3 (πœ‘ β†’ 𝐴𝑂𝐡)
2 hpg.p . . . 4 𝑃 = (Baseβ€˜πΊ)
3 hpg.d . . . 4 βˆ’ = (distβ€˜πΊ)
4 hpg.i . . . 4 𝐼 = (Itvβ€˜πΊ)
5 hpg.o . . . 4 𝑂 = {βŸ¨π‘Ž, π‘βŸ© ∣ ((π‘Ž ∈ (𝑃 βˆ– 𝐷) ∧ 𝑏 ∈ (𝑃 βˆ– 𝐷)) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (π‘ŽπΌπ‘))}
6 oppcom.a . . . 4 (πœ‘ β†’ 𝐴 ∈ 𝑃)
7 oppcom.b . . . 4 (πœ‘ β†’ 𝐡 ∈ 𝑃)
82, 3, 4, 5, 6, 7islnopp 28423 . . 3 (πœ‘ β†’ (𝐴𝑂𝐡 ↔ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡))))
91, 8mpbid 231 . 2 (πœ‘ β†’ ((Β¬ 𝐴 ∈ 𝐷 ∧ Β¬ 𝐡 ∈ 𝐷) ∧ βˆƒπ‘‘ ∈ 𝐷 𝑑 ∈ (𝐴𝐼𝐡)))
109simplrd 767 1 (πœ‘ β†’ Β¬ 𝐡 ∈ 𝐷)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆƒwrex 3069   βˆ– cdif 3945   class class class wbr 5148  {copab 5210  ran crn 5677  β€˜cfv 6543  (class class class)co 7412  Basecbs 17151  distcds 17213  TarskiGcstrkg 28111  Itvcitv 28117  LineGclng 28118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-iota 6495  df-fv 6551  df-ov 7415
This theorem is referenced by:  opphllem1  28431  opphllem2  28432  opphl  28438  lnopp2hpgb  28447  lnperpex  28487
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