Theorem tglngne 25445

Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.)

Hypotheses

Ref	Expression
tglngval.p	|- P = ( Base ` G )
tglngval.l	|- L = (LineG ` G )
tglngval.i	|- I = (Itv ` G )
tglngval.g	|- ( ph -> G e. TarskiG )
tglngval.x	|- ( ph -> X e. P )
tglngval.y	|- ( ph -> Y e. P )
tglngne.1	|- ( ph -> Z e. ( X L Y ) )

Assertion

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

Proof of Theorem tglngne

Step	Hyp	Ref	Expression
1		tglngne.1	. . . . . 6 |- ( ph -> Z e. ( X L Y ) )
2		df-ov 6653	. . . . . 6 |- ( X L Y ) = ( L ` <. X , Y >. )
3	1, 2	syl6eleq 2711	. . . . 5 |- ( ph -> Z e. ( L ` <. X , Y >. ) )
4		elfvdm 6220	. . . . 5 |- ( Z e. ( L ` <. X , Y >. ) -> <. X , Y >. e. dom L )
5	3, 4	syl 17	. . . 4 |- ( ph -> <. X , Y >. e. dom L )
6		tglngval.g	. . . . 5 |- ( ph -> G e. TarskiG )
7		tglngval.p	. . . . . 6 |- P = ( Base ` G )
8		tglngval.l	. . . . . 6 |- L = (LineG ` G )
9		tglngval.i	. . . . . 6 |- I = (Itv ` G )
10	7, 8, 9	tglnfn 25442	. . . . 5 |- ( G e. TarskiG -> L Fn ( ( P X. P ) \ _I ) )
11		fndm 5990	. . . . 5 |- ( L Fn ( ( P X. P ) \ _I ) -> dom L = ( ( P X. P ) \ _I ) )
12	6, 10, 11	3syl 18	. . . 4 |- ( ph -> dom L = ( ( P X. P ) \ _I ) )
13	5, 12	eleqtrd 2703	. . 3 |- ( ph -> <. X , Y >. e. ( ( P X. P ) \ _I ) )
14	13	eldifbd 3587	. 2 |- ( ph -> -. <. X , Y >. e. _I )
15		df-br 4654	. . . 4 |- ( X _I Y <-> <. X , Y >. e. _I )
16		tglngval.y	. . . . 5 |- ( ph -> Y e. P )
17		ideqg 5273	. . . . 5 |- ( Y e. P -> ( X _I Y <-> X = Y ) )
18	16, 17	syl 17	. . . 4 |- ( ph -> ( X _I Y <-> X = Y ) )
19	15, 18	syl5bbr 274	. . 3 |- ( ph -> ( <. X , Y >. e. _I <-> X = Y ) )
20	19	necon3bbid 2831	. 2 |- ( ph -> ( -. <. X , Y >. e. _I <-> X =/= Y ) )
21	14, 20	mpbid 222	1 |- ( ph -> X =/= Y )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   <.cop 4183   class class class wbr 4653   _I cid 5023   X. cxp 5112   dom cdm 5114   Fn wfn 5883   ` 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-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-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-trkg 25352

This theorem is referenced by:  lnhl  25510  tglnne  25523  tglineneq  25539  tglineinteq  25540  ncolncol  25541  coltr  25542  coltr3  25543  perprag  25618  opphl  25646  hlpasch  25648