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Theorem btwnlng2 28643
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)
btwnlng2.1 (𝜑𝑋 ∈ (𝑍𝐼𝑌))
Assertion
Ref Expression
btwnlng2 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng2
StepHypRef Expression
1 btwnlng2.1 . . 3 (𝜑𝑋 ∈ (𝑍𝐼𝑌))
213mix2d 1336 . 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 28576 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
122, 11mpbird 257 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1085   = wceq 1537  wcel 2106  wne 2938  cfv 6563  (class class class)co 7431  Basecbs 17245  TarskiGcstrkg 28450  Itvcitv 28456  LineGclng 28457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-trkg 28476
This theorem is referenced by:  mirln  28699  colperpexlem3  28755  outpasch  28778  hpgerlem  28788
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