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Mirrors > Home > MPE Home > Th. List > tghilberti1 | Structured version Visualization version GIF version |
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglineelsb2.1 | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
tglineelsb2.2 | ⊢ (𝜑 → 𝑄 ∈ 𝐵) |
tglineelsb2.4 | ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
Ref | Expression |
---|---|
tghilberti1 | ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tglineelsb2.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | tglineelsb2.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tglineelsb2.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
6 | tglineelsb2.2 | . . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
7 | tglineelsb2.4 | . . 3 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tgelrnln 26693 | . 2 ⊢ (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿) |
9 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx1 26696 | . 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄)) |
10 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx2 26697 | . 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄)) |
11 | eleq2 2822 | . . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑃𝐿𝑄))) | |
12 | eleq2 2822 | . . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ (𝑃𝐿𝑄))) | |
13 | 11, 12 | anbi12d 634 | . . 3 ⊢ (𝑥 = (𝑃𝐿𝑄) → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄)))) |
14 | 13 | rspcev 3530 | . 2 ⊢ (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
15 | 8, 9, 10, 14 | syl12anc 837 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∃wrex 3055 ran crn 5541 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 TarskiGcstrkg 26493 Itvcitv 26499 LineGclng 26500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 ax-un 7512 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-trkgc 26511 df-trkgb 26512 df-trkgcb 26513 df-trkg 26516 |
This theorem is referenced by: tglinethrueu 26702 |
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