Theorem btwnlng1 25514

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	|- ( ph -> G e. TarskiG )
btwnlng1.x	|- ( ph -> X e. P )
btwnlng1.y	|- ( ph -> Y e. P )
btwnlng1.z	|- ( ph -> Z e. P )
btwnlng1.d	|- ( ph -> X =/= Y )
btwnlng1.1	|- ( ph -> Z e. ( X I Y ) )

Assertion

Ref	Expression
btwnlng1	|- ( ph -> Z e. ( X L Y ) )

Proof of Theorem btwnlng1

Step	Hyp	Ref	Expression
1		btwnlng1.1	. . 3 |- ( ph -> Z e. ( X I Y ) )
2	1	3mix1d 1236	. 2 |- ( ph -> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) )
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 |- ( ph -> G e. TarskiG )
7		btwnlng1.x	. . 3 |- ( ph -> X e. P )
8		btwnlng1.y	. . 3 |- ( ph -> Y e. P )
9		btwnlng1.d	. . 3 |- ( ph -> X =/= Y )
10		btwnlng1.z	. . 3 |- ( ph -> Z e. P )
11	3, 4, 5, 6, 7, 8, 9, 10	tgellng 25448	. 2 |- ( ph -> ( Z e. ( X L Y ) <-> ( Z e. ( X I Y ) \/ X e. ( Z I Y ) \/ Y e. ( X I Z ) ) ) )
12	2, 11	mpbird 247	1 |- ( ph -> Z e. ( X L Y ) )

Syntax hints:   -> wi 4   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857  TarskiGcstrkg 25329  Itvcitv 25335  LineGclng 25336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-trkg 25352

This theorem is referenced by:  tglnne  25523  tglinerflx1  25528  tglinerflx2  25529  coltr3  25543  mirln2  25572  midexlem  25587  colperpexlem3  25624  mideulem2  25626  opphllem1  25639  opphllem2  25640  opphllem4  25642  hlpasch  25648  lnopp2hpgb  25655  colopp  25661  colhp  25662  lmieu  25676  lmimid  25686  lmiisolem  25688  hypcgrlem1  25691  hypcgrlem2  25692  trgcopyeulem  25697