Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  tghilberti1 Structured version   Visualization version   GIF version

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
 Copyright terms: Public domain W3C validator