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Theorem btwnlng1 25204
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
btwnlng1.p 𝑃 = (Base‘𝐺)
btwnlng1.i 𝐼 = (Itv‘𝐺)
btwnlng1.l 𝐿 = (LineG‘𝐺)
btwnlng1.g (𝜑𝐺 ∈ TarskiG)
btwnlng1.x (𝜑𝑋𝑃)
btwnlng1.y (𝜑𝑌𝑃)
btwnlng1.z (𝜑𝑍𝑃)
btwnlng1.d (𝜑𝑋𝑌)
btwnlng1.1 (𝜑𝑍 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
btwnlng1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng1
StepHypRef Expression
1 btwnlng1.1 . . 3 (𝜑𝑍 ∈ (𝑋𝐼𝑌))
213mix1d 1228 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
3 btwnlng1.p . . 3 𝑃 = (Base‘𝐺)
4 btwnlng1.l . . 3 𝐿 = (LineG‘𝐺)
5 btwnlng1.i . . 3 𝐼 = (Itv‘𝐺)
6 btwnlng1.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 btwnlng1.x . . 3 (𝜑𝑋𝑃)
8 btwnlng1.y . . 3 (𝜑𝑌𝑃)
9 btwnlng1.d . . 3 (𝜑𝑋𝑌)
10 btwnlng1.z . . 3 (𝜑𝑍𝑃)
113, 4, 5, 6, 7, 8, 9, 10tgellng 25138 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
122, 11mpbird 245 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1029   = wceq 1474  wcel 1938  wne 2684  cfv 5689  (class class class)co 6426  Basecbs 15579  TarskiGcstrkg 25018  Itvcitv 25024  LineGclng 25025
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-trkg 25041
This theorem is referenced by:  tglnne  25213  tglinerflx1  25218  tglinerflx2  25219  coltr3  25233  mirln2  25262  midexlem  25277  colperpexlem3  25314  mideulem2  25316  opphllem1  25329  opphllem2  25330  opphllem4  25332  hlpasch  25338  lnopp2hpgb  25345  colopp  25351  colhp  25352  lmieu  25366  lmimid  25376  lmiisolem  25378  hypcgrlem1  25381  hypcgrlem2  25382  trgcopyeulem  25387
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