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| Description: The "opposite to a line" relation is irreflexive. (Contributed by Thierry Arnoux, 4-Mar-2020.) | 
| 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) | 
| Ref | Expression | 
|---|---|
| oppnid | ⊢ (𝜑 → ¬ 𝐴𝑂𝐴) | 
| 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 730 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐺 ∈ TarskiG) | 
| 6 | oppnid.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 7 | 6 | ad3antrrr 730 | . . . . 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 730 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐷 ∈ ran 𝐿) | 
| 11 | simplr 768 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝐷) | |
| 12 | 1, 8, 3, 5, 10, 11 | tglnpt 28558 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝑃) | 
| 13 | simpr 484 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴)) | |
| 14 | 1, 2, 3, 5, 7, 12, 13 | axtgbtwnid 28475 | . . . 4 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡) | 
| 15 | 14, 11 | eqeltrd 2840 | . . 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 28748 | . . . 4 ⊢ (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴)))) | 
| 18 | 17 | simplbda 499 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴)) | 
| 19 | 15, 18 | r19.29a 3161 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → 𝐴 ∈ 𝐷) | 
| 20 | 17 | simprbda 498 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷)) | 
| 21 | 20 | simpld 494 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ¬ 𝐴 ∈ 𝐷) | 
| 22 | 19, 21 | pm2.65da 816 | 1 ⊢ (𝜑 → ¬ 𝐴𝑂𝐴) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ∖ cdif 3947 class class class wbr 5142 {copab 5204 ran crn 5685 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 distcds 17307 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-iota 6513 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-trkgb 28458 df-trkg 28462 | 
| This theorem is referenced by: lnoppnhpg 28773 | 
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