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Mirrors > Home > MPE Home > Th. List > tglnne0 | Structured version Visualization version GIF version |
Description: A line 𝐴 has at least one point. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
tglnne0.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglnne0.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglnne0.1 | ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
Ref | Expression |
---|---|
tglnne0 | ⊢ (𝜑 → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2798 | . . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
3 | tglnne0.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglnne0.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad3antrrr 729 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | simpllr 775 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (Base‘𝐺)) | |
7 | simplr 768 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (Base‘𝐺)) | |
8 | simprr 772 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
9 | 1, 2, 3, 5, 6, 7, 8 | tglinerflx1 26427 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦)) |
10 | simprl 770 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = (𝑥𝐿𝑦)) | |
11 | 9, 10 | eleqtrrd 2893 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐴) |
12 | 11 | ne0d 4251 | . 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 ≠ ∅) |
13 | tglnne0.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
14 | 1, 2, 3, 4, 13 | tgisline 26421 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
15 | 12, 14 | r19.29vva 3292 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∅c0 4243 ran crn 5520 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: hpgerlem 26559 |
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