Theorem tglnssp

Description: Lines are subset of the geometry base set. That is, lines are sets of points. (Contributed by Thierry Arnoux, 17-May-2019)

	Ref	Expression
Hypotheses	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 )
	tglngval.z	|- ( ph -> X =/= Y )
Assertion	tglnssp	|- ( ph -> ( X L Y ) C_ P )

Proof

Step	Hyp	Ref	Expression
1		tglngval.p	|- P = ( Base ` G )
2		tglngval.l	|- L = ( LineG ` G )
3		tglngval.i	|- I = ( Itv ` G )
4		tglngval.g	|- ( ph -> G e. TarskiG )
5		tglngval.x	|- ( ph -> X e. P )
6		tglngval.y	|- ( ph -> Y e. P )
7		tglngval.z	|- ( ph -> X =/= Y )
8	1 2 3 4 5 6 7	tglngval	|- ( ph -> ( X L Y ) = { z e. P | ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) } )
9		ssrab2	|- { z e. P | ( z e. ( X I Y ) \/ X e. ( z I Y ) \/ Y e. ( X I z ) ) } C_ P
10	8 9	eqsstrdi	|- ( ph -> ( X L Y ) C_ P )