Theorem tglnne 25523

Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-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 )
tglnne.x	|- ( ph -> X e. B )
tglnne.y	|- ( ph -> Y e. B )
tglnne.1	|- ( ph -> ( X L Y ) e. ran L )

Assertion

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

Proof of Theorem tglnne

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tglineelsb2.p	. . 3 |- B = ( Base ` G )
2		tglineelsb2.l	. . 3 |- L = (LineG ` G )
3		tglineelsb2.i	. . 3 |- I = (Itv ` G )
4		tglineelsb2.g	. . . 4 |- ( ph -> G e. TarskiG )
5	4	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> G e. TarskiG )
6		tglnne.x	. . . 4 |- ( ph -> X e. B )
7	6	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> X e. B )
8		tglnne.y	. . . 4 |- ( ph -> Y e. B )
9	8	ad3antrrr 766	. . 3 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> Y e. B )
10		simpllr 799	. . . . 5 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> x e. B )
11		simplr 792	. . . . 5 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> y e. B )
12		simprr 796	. . . . 5 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> x =/= y )
13		eqid 2622	. . . . . 6 |- ( dist ` G ) = ( dist ` G )
14	1, 13, 3, 5, 10, 11	tgbtwntriv1 25386	. . . . 5 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> x e. ( x I y ) )
15	1, 3, 2, 5, 10, 11, 10, 12, 14	btwnlng1 25514	. . . 4 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> x e. ( x L y ) )
16		simprl 794	. . . 4 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> ( X L Y ) = ( x L y ) )
17	15, 16	eleqtrrd 2704	. . 3 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> x e. ( X L Y ) )
18	1, 2, 3, 5, 7, 9, 17	tglngne 25445	. 2 |- ( ( ( ( ph /\ x e. B ) /\ y e. B ) /\ ( ( X L Y ) = ( x L y ) /\ x =/= y ) ) -> X =/= Y )
19		tglnne.1	. . 3 |- ( ph -> ( X L Y ) e. ran L )
20	1, 3, 2, 4, 19	tgisline 25522	. 2 |- ( ph -> E. x e. B E. y e. B ( ( X L Y ) = ( x L y ) /\ x =/= y ) )
21	18, 20	r19.29vva 3081	1 |- ( ph -> X =/= Y )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ran crn 5115   ` 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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949

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-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-trkgc 25347  df-trkgb 25348  df-trkgcb 25349  df-trkg 25352

This theorem is referenced by:  footne  25615  footeq  25616  hlperpnel  25617  colperp  25621  mideulem2  25626  opphllem  25627  midex  25629  opphllem3  25641  opphllem6  25644  opphl  25646  lmieu  25676  lnperpex  25695  trgcopy  25696