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Mirrors > Home > MPE Home > Th. List > oppnid | Structured version Visualization version GIF version |
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 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 26335 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ 𝑃) |
13 | simpr 487 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝑡 ∈ (𝐴𝐼𝐴)) | |
14 | 1, 2, 3, 5, 7, 12, 13 | axtgbtwnid 26252 | . . . 4 ⊢ ((((𝜑 ∧ 𝐴𝑂𝐴) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐴)) → 𝐴 = 𝑡) |
15 | 14, 11 | eqeltrd 2913 | . . 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 26525 | . . . 4 ⊢ (𝜑 → (𝐴𝑂𝐴 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴)))) |
18 | 17 | simplbda 502 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐴)) |
19 | 15, 18 | r19.29a 3289 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → 𝐴 ∈ 𝐷) |
20 | 17 | simprbda 501 | . . 3 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → (¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐴 ∈ 𝐷)) |
21 | 20 | simpld 497 | . 2 ⊢ ((𝜑 ∧ 𝐴𝑂𝐴) → ¬ 𝐴 ∈ 𝐷) |
22 | 19, 21 | pm2.65da 815 | 1 ⊢ (𝜑 → ¬ 𝐴𝑂𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 ∖ cdif 3933 class class class wbr 5066 {copab 5128 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 distcds 16574 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-iota 6314 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-trkgb 26235 df-trkg 26239 |
This theorem is referenced by: lnoppnhpg 26550 |
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