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Theorem tghilberti1 26431
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 26424 . 2 (𝜑 → (𝑃𝐿𝑄) ∈ ran 𝐿)
91, 2, 3, 4, 5, 6, 7tglinerflx1 26427 . 2 (𝜑𝑃 ∈ (𝑃𝐿𝑄))
101, 2, 3, 4, 5, 6, 7tglinerflx2 26428 . 2 (𝜑𝑄 ∈ (𝑃𝐿𝑄))
11 eleq2 2878 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑃𝑥𝑃 ∈ (𝑃𝐿𝑄)))
12 eleq2 2878 . . . 4 (𝑥 = (𝑃𝐿𝑄) → (𝑄𝑥𝑄 ∈ (𝑃𝐿𝑄)))
1311, 12anbi12d 633 . . 3 (𝑥 = (𝑃𝐿𝑄) → ((𝑃𝑥𝑄𝑥) ↔ (𝑃 ∈ (𝑃𝐿𝑄) ∧ 𝑄 ∈ (𝑃𝐿𝑄))))
1413rspcev 3571 . 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 399   = wceq 1538  wcel 2111  wne 2987  wrex 3107  ran crn 5520  cfv 6324  (class class class)co 7135  Basecbs 16475  TarskiGcstrkg 26224  Itvcitv 26230  LineGclng 26231
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-trkgc 26242  df-trkgb 26243  df-trkgcb 26244  df-trkg 26247
This theorem is referenced by:  tglinethrueu  26433
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