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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 26416 | . 2 ⊢ (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿) |
9 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx1 26419 | . 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄)) |
10 | 1, 2, 3, 4, 5, 6, 7 | tglinerflx2 26420 | . 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄)) |
11 | eleq2 2901 | . . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑃𝐿𝑄))) | |
12 | eleq2 2901 | . . . 4 ⊢ (𝑥 = (𝑃𝐿𝑄) → (𝑄 ∈ 𝑥 ↔ 𝑄 ∈ (𝑃𝐿𝑄))) | |
13 | 11, 12 | anbi12d 632 | . . 3 ⊢ (𝑥 = (𝑃𝐿𝑄) → ((𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄)))) |
14 | 13 | rspcev 3623 | . 2 ⊢ (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
15 | 8, 9, 10, 14 | syl12anc 834 | 1 ⊢ (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃 ∈ 𝑥 ∧ 𝑄 ∈ 𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ran crn 5556 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 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-pow 5266 ax-pr 5330 ax-un 7461 |
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-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 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-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-trkgc 26234 df-trkgb 26235 df-trkgcb 26236 df-trkg 26239 |
This theorem is referenced by: tglinethrueu 26425 |
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