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Theorem btwnlng3 26413
 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)
btwnlng3.1 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
Assertion
Ref Expression
btwnlng3 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng3
StepHypRef Expression
1 btwnlng3.1 . . 3 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
213mix3d 1335 . 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 26345 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
122, 11mpbird 260 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ w3o 1083   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ‘cfv 6334  (class class class)co 7140  Basecbs 16474  TarskiGcstrkg 26222  Itvcitv 26228  LineGclng 26229 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307 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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-trkg 26245 This theorem is referenced by:  midexlem  26484  footexALT  26510  footexlem1  26511  footexlem2  26512  mideulem2  26526  opphllem1  26539  outpasch  26547  colhp  26562
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