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Theorem btwncolg3 28583
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 1338 . 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 28580 . 2 (𝜑 → ((𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
112, 10mpbird 257 1 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ∨ 𝑋 = 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 846  w3o 1086   = wceq 1537  wcel 2108  cfv 6573  (class class class)co 7448  Basecbs 17258  TarskiGcstrkg 28453  Itvcitv 28459  LineGclng 28460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-trkgc 28474  df-trkgcb 28476  df-trkg 28479
This theorem is referenced by:  tgdim01ln  28590  lnxfr  28592  tgidinside  28597  tgbtwnconn1lem3  28600  tgbtwnconnln3  28604  tgbtwnconnln1  28606  tgbtwnconnln2  28607  legov  28611  legov2  28612  legtrd  28615  tglineeltr  28657  krippenlem  28716  midexlem  28718  footexALT  28744  footexlem2  28746  mideulem2  28760  hlpasch  28782  hypcgrlem1  28825  cgracol  28854
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