Theorem oppnid 27994

Description: The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-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)
oppnid.1	⊢ (𝜑 → 𝐴 ∈ 𝑃)

Assertion

Ref	Expression
oppnid	⊢ (𝜑 → ¬ 𝐴𝑂𝐴)

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

Proof of Theorem oppnid

Step	Hyp	Ref	Expression
1		hpg.p	. . . . 5 ⊢ 𝑃 = (Base‘𝐺)
2		hpg.d	. . . . 5 ⊢ − = (dist‘𝐺)
3		hpg.i	. . . . 5 ⊢ 𝐼 = (Itv‘𝐺)
4		opphl.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG)
6		oppnid.1	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7	6	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 ∈ 𝑃)
8		opphl.l	. . . . . 6 ⊢ 𝐿 = (LineG‘𝐺)
9		opphl.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿)
10	9	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿)
11		simplr 767	. . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝐷)
12	1, 8, 3, 5, 10, 11	tglnpt 27797	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝑃)
13		simpr 485	. . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴))
14	1, 2, 3, 5, 7, 12, 13	axtgbtwnid 27714	. . . 4 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡)
15	14, 11	eqeltrd 2833	. . 3 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 ∈ 𝐷)
16		hpg.o	. . . . 5 ⊢ 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
17	1, 2, 3, 16, 6, 6	islnopp 27987	. . . 4 ⊢ (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))))
18	17	simplbda 500	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴))
19	15, 18	r19.29a 3162	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → 𝐴 ∈ 𝐷)
20	17	simprbda 499	. . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷))
21	20	simpld 495	. 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ¬ 𝐴 ∈ 𝐷)
22	19, 21	pm2.65da 815	1 ⊢ (𝜑 → ¬ 𝐴𝑂𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070   ∖ cdif 3945   class class class wbr 5148  {copab 5210  ran crn 5677  ‘cfv 6543  (class class class)co 7408  Basecbs 17143  distcds 17205  TarskiGcstrkg 27675  Itvcitv 27681  LineGclng 27682

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-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-cnv 5684  df-dm 5686  df-rn 5687  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-trkgb 27697  df-trkg 27701

This theorem is referenced by:  lnoppnhpg  28012