Theorem hleqnid

Description: The endpoint does not belong to the half-line. (Contributed by Thierry Arnoux, 3-Mar-2020)

	Ref	Expression
Hypotheses	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 )
	hlln.1	|- ( ph -> G e. TarskiG )
Assertion	hleqnid	|- ( ph -> -. A ( K ` A ) B )

Proof

Step	Hyp	Ref	Expression
1		ishlg.p	|- P = ( Base ` G )
2		ishlg.i	|- I = ( Itv ` G )
3		ishlg.k	|- K = ( hlG ` G )
4		ishlg.a	|- ( ph -> A e. P )
5		ishlg.b	|- ( ph -> B e. P )
6		ishlg.c	|- ( ph -> C e. P )
7		hlln.1	|- ( ph -> G e. TarskiG )
8		neirr	|- -. A =/= A
9	8	a1i	|- ( ph -> -. A =/= A )
10	4	adantr	|- ( ( ph /\ A ( K ` A ) B ) -> A e. P )
11	5	adantr	|- ( ( ph /\ A ( K ` A ) B ) -> B e. P )
12	7	adantr	|- ( ( ph /\ A ( K ` A ) B ) -> G e. TarskiG )
13		simpr	|- ( ( ph /\ A ( K ` A ) B ) -> A ( K ` A ) B )
14	1 2 3 10 11 10 12 13	hlne1	|- ( ( ph /\ A ( K ` A ) B ) -> A =/= A )
15	9 14	mtand	|- ( ph -> -. A ( K ` A ) B )