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Theorem btwnlng1 25508
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)
btwnlng1.1 (𝜑𝑍 ∈ (𝑋𝐼𝑌))
Assertion
Ref Expression
btwnlng1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))

Proof of Theorem btwnlng1
StepHypRef Expression
1 btwnlng1.1 . . 3 (𝜑𝑍 ∈ (𝑋𝐼𝑌))
213mix1d 1235 . 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 25442 . 2 (𝜑 → (𝑍 ∈ (𝑋𝐿𝑌) ↔ (𝑍 ∈ (𝑋𝐼𝑌) ∨ 𝑋 ∈ (𝑍𝐼𝑌) ∨ 𝑌 ∈ (𝑋𝐼𝑍))))
122, 11mpbird 247 1 (𝜑𝑍 ∈ (𝑋𝐿𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3o 1036   = wceq 1482  wcel 1989  wne 2793  cfv 5886  (class class class)co 6647  Basecbs 15851  TarskiGcstrkg 25323  Itvcitv 25329  LineGclng 25330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-trkg 25346
This theorem is referenced by:  tglnne  25517  tglinerflx1  25522  tglinerflx2  25523  coltr3  25537  mirln2  25566  midexlem  25581  colperpexlem3  25618  mideulem2  25620  opphllem1  25633  opphllem2  25634  opphllem4  25636  hlpasch  25642  lnopp2hpgb  25649  colopp  25655  colhp  25656  lmieu  25670  lmimid  25680  lmiisolem  25682  hypcgrlem1  25685  hypcgrlem2  25686  trgcopyeulem  25691
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