Theorem tgbtwnne 25385

Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.)

Hypotheses

Ref	Expression
tkgeom.p	|- P = ( Base ` G )
tkgeom.d	|- .- = ( dist ` G )
tkgeom.i	|- I = (Itv ` G )
tkgeom.g	|- ( ph -> G e. TarskiG )
tgbtwntriv2.1	|- ( ph -> A e. P )
tgbtwntriv2.2	|- ( ph -> B e. P )
tgbtwncomb.3	|- ( ph -> C e. P )
tgbtwnne.1	|- ( ph -> B e. ( A I C ) )
tgbtwnne.2	|- ( ph -> B =/= A )

Assertion

Ref	Expression
tgbtwnne	|- ( ph -> A =/= C )

Proof of Theorem tgbtwnne

Step	Hyp	Ref	Expression
1		tkgeom.p	. . . . 5 |- P = ( Base ` G )
2		tkgeom.d	. . . . 5 |- .- = ( dist ` G )
3		tkgeom.i	. . . . 5 |- I = (Itv ` G )
4		tkgeom.g	. . . . . 6 |- ( ph -> G e. TarskiG )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ A = C ) -> G e. TarskiG )
6		tgbtwntriv2.1	. . . . . 6 |- ( ph -> A e. P )
7	6	adantr 481	. . . . 5 |- ( ( ph /\ A = C ) -> A e. P )
8		tgbtwntriv2.2	. . . . . 6 |- ( ph -> B e. P )
9	8	adantr 481	. . . . 5 |- ( ( ph /\ A = C ) -> B e. P )
10		tgbtwnne.1	. . . . . . 7 |- ( ph -> B e. ( A I C ) )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ A = C ) -> B e. ( A I C ) )
12		simpr 477	. . . . . . 7 |- ( ( ph /\ A = C ) -> A = C )
13	12	oveq2d 6666	. . . . . 6 |- ( ( ph /\ A = C ) -> ( A I A ) = ( A I C ) )
14	11, 13	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ A = C ) -> B e. ( A I A ) )
15	1, 2, 3, 5, 7, 9, 14	axtgbtwnid 25365	. . . 4 |- ( ( ph /\ A = C ) -> A = B )
16	15	eqcomd 2628	. . 3 |- ( ( ph /\ A = C ) -> B = A )
17		tgbtwnne.2	. . . . 5 |- ( ph -> B =/= A )
18	17	adantr 481	. . . 4 |- ( ( ph /\ A = C ) -> B =/= A )
19	18	neneqd 2799	. . 3 |- ( ( ph /\ A = C ) -> -. B = A )
20	16, 19	pm2.65da 600	. 2 |- ( ph -> -. A = C )
21	20	neqned 2801	1 |- ( ph -> A =/= C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335

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-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  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-iota 5851  df-fv 5896  df-ov 6653  df-trkgb 25348  df-trkg 25352

This theorem is referenced by:  mideulem2  25626  opphllem  25627  outpasch  25647  lnopp2hpgb  25655  lmieu  25676  dfcgra2  25721