Metamath Proof Explorer


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 ) )