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Theorem tghilberti1 26350
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 26343 . 2 (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
91, 2, 3, 4, 5, 6, 7tglinerflx1 26346 . 2 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
101, 2, 3, 4, 5, 6, 7tglinerflx2 26347 . 2 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
11 eleq2 2898 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑃𝑥𝑃 ∈ (𝑃𝐿𝑄)))
12 eleq2 2898 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑄𝑥𝑄 ∈ (𝑃𝐿𝑄)))
1311, 12anbi12d 630 . . 3 (𝑥 = (𝑃𝐿𝑄) → ((𝑃𝑥𝑄𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
1413rspcev 3620 . 2 (((𝑃𝐿𝑄) ∈ ran 𝐿 ∧ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))) → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
158, 9, 10, 14syl12anc 832 1 (𝜑 → ∃𝑥 ∈ ran 𝐿(𝑃𝑥𝑄𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wne 3013  wrex 3136  ran crn 5549  cfv 6348  (class class class)co 7145  Basecbs 16471  TarskiGcstrkg 26143  Itvcitv 26149  LineGclng 26150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-trkgc 26161  df-trkgb 26162  df-trkgcb 26163  df-trkg 26166
This theorem is referenced by:  tglinethrueu  26352
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