Theorem ncolne1 25520

Description: Non-colinear points are different. (Contributed by Thierry Arnoux, 8-Aug-2019.)

Hypotheses

Ref	Expression
tglineelsb2.p	|- B = ( Base ` G )
tglineelsb2.i	|- I = (Itv ` G )
tglineelsb2.l	|- L = (LineG ` G )
tglineelsb2.g	|- ( ph -> G e. TarskiG )
ncolne.x	|- ( ph -> X e. B )
ncolne.y	|- ( ph -> Y e. B )
ncolne.z	|- ( ph -> Z e. B )
ncolne.2	|- ( ph -> -. ( X e. ( Y L Z ) \/ Y = Z ) )

Assertion

Ref	Expression
ncolne1	|- ( ph -> X =/= Y )

Proof of Theorem ncolne1

Step	Hyp	Ref	Expression
1		ncolne.2	. . 3 |- ( ph -> -. ( X e. ( Y L Z ) \/ Y = Z ) )
2		tglineelsb2.p	. . . 4 |- B = ( Base ` G )
3		tglineelsb2.l	. . . 4 |- L = (LineG ` G )
4		tglineelsb2.i	. . . 4 |- I = (Itv ` G )
5		tglineelsb2.g	. . . . 5 |- ( ph -> G e. TarskiG )
6	5	adantr 481	. . . 4 |- ( ( ph /\ X = Y ) -> G e. TarskiG )
7		ncolne.y	. . . . 5 |- ( ph -> Y e. B )
8	7	adantr 481	. . . 4 |- ( ( ph /\ X = Y ) -> Y e. B )
9		ncolne.z	. . . . 5 |- ( ph -> Z e. B )
10	9	adantr 481	. . . 4 |- ( ( ph /\ X = Y ) -> Z e. B )
11		ncolne.x	. . . . 5 |- ( ph -> X e. B )
12	11	adantr 481	. . . 4 |- ( ( ph /\ X = Y ) -> X e. B )
13		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
14	2, 13, 4, 6, 12, 10	tgbtwntriv1 25386	. . . . 5 |- ( ( ph /\ X = Y ) -> X e. ( X I Z ) )
15		simpr 477	. . . . . 6 |- ( ( ph /\ X = Y ) -> X = Y )
16	15	oveq1d 6665	. . . . 5 |- ( ( ph /\ X = Y ) -> ( X I Z ) = ( Y I Z ) )
17	14, 16	eleqtrd 2703	. . . 4 |- ( ( ph /\ X = Y ) -> X e. ( Y I Z ) )
18	2, 3, 4, 6, 8, 10, 12, 17	btwncolg1 25450	. . 3 |- ( ( ph /\ X = Y ) -> ( X e. ( Y L Z ) \/ Y = Z ) )
19	1, 18	mtand 691	. 2 |- ( ph -> -. X = Y )
20	19	neqned 2801	1 |- ( ph -> X =/= Y )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  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-pw 4160  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-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  ncolne2  25521  tglineneq  25539  midexlem  25587  mideulem2  25626  outpasch  25647  hlpasch  25648  trgcopy  25696  trgcopyeulem  25697  acopy  25724  acopyeu  25725  cgrg3col4  25734  tgasa1  25739  isoas  25744