Theorem otpsle

Description: The order of a topological ordered space. (Contributed by Mario Carneiro, 12-Nov-2015) (Revised by AV, 9-Sep-2021)

	Ref	Expression
Hypothesis	otpsstr.w	|- K = { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
Assertion	otpsle	|- ( .<_ e. V -> .<_ = ( le ` K ) )

Proof

Step	Hyp	Ref	Expression
1		otpsstr.w	|- K = { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
2	1	otpsstr	|- K Struct <. 1 , ; 1 0 >.
3		pleid	|- le = Slot ( le ` ndx )
4		snsstp3	|- { <. ( le ` ndx ) , .<_ >. } C_ { <. ( Base ` ndx ) , B >. , <. ( TopSet ` ndx ) , J >. , <. ( le ` ndx ) , .<_ >. }
5	4 1	sseqtrri	|- { <. ( le ` ndx ) , .<_ >. } C_ K
6	2 3 5	strfv	|- ( .<_ e. V -> .<_ = ( le ` K ) )