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Theorem btwnlng2 25931
 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 1442 . 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 25864 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
122, 11mpbird 249 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1112   = wceq 1658   ∈ wcel 2166   ≠ wne 2998  ‘cfv 6122  (class class class)co 6904  Basecbs 16221  TarskiGcstrkg 25741  Itvcitv 25747  LineGclng 25748 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-iota 6085  df-fun 6124  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-trkg 25764 This theorem is referenced by:  mirln  25987  colperpexlem3  26040  outpasch  26063  hpgerlem  26073
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