Theorem hlne2 25501

Description: The half-line relation implies inequality. (Contributed by Thierry Arnoux, 22-Feb-2020.)

Hypotheses

Ref	Expression
ishlg.p	|- P = ( Base ` G )
ishlg.i	|- I = (Itv ` G )
ishlg.k	|- K = (hlG ` G )
ishlg.a	|- ( ph -> A e. P )
ishlg.b	|- ( ph -> B e. P )
ishlg.c	|- ( ph -> C e. P )
ishlg.g	|- ( ph -> G e. V )
hlcomd.1	|- ( ph -> A ( K ` C ) B )

Assertion

Ref	Expression
hlne2	|- ( ph -> B =/= C )

Proof of Theorem hlne2

Step	Hyp	Ref	Expression
1		hlcomd.1	. . 3 |- ( ph -> A ( K ` C ) B )
2		ishlg.p	. . . 4 |- P = ( Base ` G )
3		ishlg.i	. . . 4 |- I = (Itv ` G )
4		ishlg.k	. . . 4 |- K = (hlG ` G )
5		ishlg.a	. . . 4 |- ( ph -> A e. P )
6		ishlg.b	. . . 4 |- ( ph -> B e. P )
7		ishlg.c	. . . 4 |- ( ph -> C e. P )
8		ishlg.g	. . . 4 |- ( ph -> G e. V )
9	2, 3, 4, 5, 6, 7, 8	ishlg 25497	. . 3 |- ( ph -> ( A ( K ` C ) B <-> ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) ) )
10	1, 9	mpbid 222	. 2 |- ( ph -> ( A =/= C /\ B =/= C /\ ( A e. ( C I B ) \/ B e. ( C I A ) ) ) )
11	10	simp2d 1074	1 |- ( ph -> B =/= C )

Syntax hints:   -> wi 4   \/ wo 383   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857  Itvcitv 25335  hlGchlg 25495

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-rep 4771  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-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-reu 2919  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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-hlg 25496

This theorem is referenced by:  hltr  25505  opphllem5  25643  hlpasch  25648  colhp  25662  iscgra1  25702  cgrane3  25706  cgrane4  25707  inaghl  25731