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Theorem btwncolg3 26357
Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019.)
Hypotheses
Ref Expression
tglngval.p 𝑃 = (Base‘𝐺)
tglngval.l 𝐿 = (LineG‘𝐺)
tglngval.i 𝐼 = (Itv‘𝐺)
tglngval.g (𝜑𝐺 ∈ TarskiG)
tglngval.x (𝜑𝑋𝑃)
tglngval.y (𝜑𝑌𝑃)
tgcolg.z (𝜑𝑍𝑃)
btwncolg3.z (𝜑𝑌 ∈ (𝑋𝐼𝑍))
Assertion
Ref Expression
btwncolg3 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))

Proof of Theorem btwncolg3
StepHypRef Expression
1 btwncolg3.z . . 3 (𝜑𝑌 ∈ (𝑋𝐼𝑍))
213mix3d 1335 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍)))
3 tglngval.p . . 3 𝑃 = (Base‘𝐺)
4 tglngval.l . . 3 𝐿 = (LineG‘𝐺)
5 tglngval.i . . 3 𝐼 = (Itv‘𝐺)
6 tglngval.g . . 3 (𝜑𝐺 ∈ TarskiG)
7 tglngval.x . . 3 (𝜑𝑋𝑃)
8 tglngval.y . . 3 (𝜑𝑌𝑃)
9 tgcolg.z . . 3 (𝜑𝑍𝑃)
103, 4, 5, 6, 7, 8, 9tgcolg 26354 . 2 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
112, 10mpbird 260 1 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844  w3o 1083   = wceq 1538  wcel 2115  cfv 6344  (class class class)co 7150  Basecbs 16486  TarskiGcstrkg 26230  Itvcitv 26236  LineGclng 26237
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3483  df-sbc 3760  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-nul 4278  df-if 4452  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7153  df-oprab 7154  df-mpo 7155  df-trkgc 26248  df-trkgcb 26250  df-trkg 26253
This theorem is referenced by:  tgdim01ln  26364  lnxfr  26366  tgidinside  26371  tgbtwnconn1lem3  26374  tgbtwnconnln3  26378  tgbtwnconnln1  26380  tgbtwnconnln2  26381  legov  26385  legov2  26386  legtrd  26389  tglineeltr  26431  krippenlem  26490  midexlem  26492  footexALT  26518  footexlem2  26520  mideulem2  26534  hlpasch  26556  hypcgrlem1  26599  cgracol  26628
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