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Theorem tghilberti1 27753
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 27746 . 2 (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
91, 2, 3, 4, 5, 6, 7tglinerflx1 27749 . 2 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
101, 2, 3, 4, 5, 6, 7tglinerflx2 27750 . 2 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
11 eleq2 2821 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑃𝑥𝑃 ∈ (𝑃𝐿𝑄)))
12 eleq2 2821 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑄𝑥𝑄 ∈ (𝑃𝐿𝑄)))
1311, 12anbi12d 631 . . 3 (𝑥 = (𝑃𝐿𝑄) → ((𝑃𝑥𝑄𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
1413rspcev 3609 . 2 (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
158, 9, 10, 14syl12anc 835 1 (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  wne 2939  wrex 3069  ran crn 5670  cfv 6532  (class class class)co 7393  Basecbs 17126  TarskiGcstrkg 27543  Itvcitv 27549  LineGclng 27550
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-trkgc 27564  df-trkgb 27565  df-trkgcb 27566  df-trkg 27569
This theorem is referenced by:  tglinethrueu  27755
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