Theorem oppne2 27855

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
oppne2	⊢ (𝜑 → ¬ 𝐵 ∈ 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 oppne2

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 27852	. . 3 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))))
9	1, 8	mpbid 231	. 2 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))
10	9	simplrd 768	1 ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐷)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3069   ∖ cdif 3940   class class class wbr 5140  {copab 5202  ran crn 5669  ‘cfv 6531  (class class class)co 7392  Basecbs 17125  distcds 17187  TarskiGcstrkg 27540  Itvcitv 27546  LineGclng 27547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-opab 5203  df-iota 6483  df-fv 6539  df-ov 7395

This theorem is referenced by:  opphllem1  27860  opphllem2  27861  opphl  27867  lnopp2hpgb  27876  lnperpex  27916