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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 2823 | . . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2823 | . . . . 5 ⊢ (Itv‘𝐺) = (Itv‘𝐺) | |
3 | tglnne0.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglnne0.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad3antrrr 728 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | simpllr 774 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (Base‘𝐺)) | |
7 | simplr 767 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ (Base‘𝐺)) | |
8 | simprr 771 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
9 | 1, 2, 3, 5, 6, 7, 8 | tglinerflx1 26421 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦)) |
10 | simprl 769 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐴 = (𝑥𝐿𝑦)) | |
11 | 9, 10 | eleqtrrd 2918 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) ∧ 𝑦 ∈ (Base‘𝐺)) ∧ (𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐴) |
12 | 11 | ne0d 4303 | . 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 26415 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝐴 = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
15 | 12, 14 | r19.29vva 3338 | 1 ⊢ (𝜑 → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 ran crn 5558 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-trkgc 26236 df-trkgb 26237 df-trkgcb 26238 df-trkg 26241 |
This theorem is referenced by: hpgerlem 26553 |
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