Theorem oppne1 29094

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 = (Base‘𝐺)
hpg.d	⊢ − = (dist‘𝐺)
hpg.i	⊢ 𝐼 = (Itv‘𝐺)
hpg.o	⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
opphl.l	⊢ 𝐿 = (LineG‘𝐺)
opphl.d	⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
opphl.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
oppcom.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
oppcom.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
oppcom.o	⊢ (𝜑 → 𝐴𝑂𝐵)

Assertion

Ref	Expression
oppne1	⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)

Distinct variable groups:   𝐷,𝑎,𝑏   𝐼,𝑎,𝑏   𝑃,𝑎,𝑏   𝑡,𝐴   𝑡,𝐵   𝑡,𝐷   𝑡,𝐺   𝑡,𝐿   𝑡,𝐼   𝑡,𝑂   𝑡,𝑃   𝜑,𝑡   𝑡, −   𝑡,𝑎,𝑏

Allowed substitution hints:   𝜑(𝑎, 𝑏)   𝐴(𝑎, 𝑏)   𝐵(𝑎, 𝑏)   𝐺(𝑎, 𝑏)   𝐿(𝑎, 𝑏)   − (𝑎, 𝑏)   𝑂(𝑎, 𝑏)

Proof of Theorem oppne1

Step	Hyp	Ref	Expression
1		oppcom.o	. . 3 ⊢ (𝜑 → 𝐴𝑂𝐵)
2		hpg.p	. . . 4 ⊢ 𝑃 = (Base‘𝐺)
3		hpg.d	. . . 4 ⊢ − = (dist‘𝐺)
4		hpg.i	. . . 4 ⊢ 𝐼 = (Itv‘𝐺)
5		hpg.o	. . . 4 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
6		oppcom.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		oppcom.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	2, 3, 4, 5, 6, 7	islnopp 29092	. . 3 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
9	1, 8	mpbid 235	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
10	9	simplld 780	1 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3088   ∖ cdif 3899   class class class wbr 5107  {copab 5171  ran crn 5660  ‘cfv 6537  (class class class)co 7416  Basecbs 17305  distcds 17355  TarskiGcstrkg 28766  Itvcitv 28772  LineGclng 28773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-iota 6493  df-fv 6545  df-ov 7419

This theorem is used by:  oppne3  29096  opptgdim2  29098  opphllem1  29100  opphllem2  29101  opphllem4  29103  opphl  29107  hpgne1  29116  hpgne2  29117  lnopp2hpgb  29118  tgaaddcpbllem1  29226  tgaaddcpbllem2  29227  tgaaddcpbl  29229  tgaltai  29310