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Theorem tghilberti1 25753
Description: There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypotheses
Ref Expression
tglineelsb2.p 𝐵 = (Base‘𝐺)
tglineelsb2.i 𝐼 = (Itv‘𝐺)
tglineelsb2.l 𝐿 = (LineG‘𝐺)
tglineelsb2.g (𝜑𝐺 ∈ TarskiG)
tglineelsb2.1 (𝜑𝑃𝐵)
tglineelsb2.2 (𝜑𝑄𝐵)
tglineelsb2.4 (𝜑𝑃𝑄)
Assertion
Ref Expression
tghilberti1 (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥,𝐼   𝑥,𝐿   𝑥,𝑃   𝑥,𝑄   𝜑,𝑥

Proof of Theorem tghilberti1
StepHypRef 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 (𝜑𝐺 ∈ TarskiG)
5 tglineelsb2.1 . . 3 (𝜑𝑃𝐵)
6 tglineelsb2.2 . . 3 (𝜑𝑄𝐵)
7 tglineelsb2.4 . . 3 (𝜑𝑃𝑄)
81, 2, 3, 4, 5, 6, 7tgelrnln 25746 . 2 (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
91, 2, 3, 4, 5, 6, 7tglinerflx1 25749 . 2 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
101, 2, 3, 4, 5, 6, 7tglinerflx2 25750 . 2 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
11 eleq2 2839 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑃𝑥𝑃 ∈ (𝑃𝐿𝑄)))
12 eleq2 2839 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑄𝑥𝑄 ∈ (𝑃𝐿𝑄)))
1311, 12anbi12d 616 . . 3 (𝑥 = (𝑃𝐿𝑄) → ((𝑃𝑥𝑄𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
1413rspcev 3460 . 2 (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
158, 9, 10, 14syl12anc 1474 1 (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  wrex 3062  ran crn 5250  cfv 6031  (class class class)co 6793  Basecbs 16064  TarskiGcstrkg 25550  Itvcitv 25556  LineGclng 25557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-trkgc 25568  df-trkgb 25569  df-trkgcb 25570  df-trkg 25573
This theorem is referenced by:  tglinethrueu  25755
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