Metamath Proof Explorer


Theorem supsn

Description: The supremum of a singleton. (Contributed by NM, 2-Oct-2007)

Ref Expression
Assertion supsn
|- ( ( R Or A /\ B e. A ) -> sup ( { B } , A , R ) = B )

Proof

Step Hyp Ref Expression
1 dfsn2
 |-  { B } = { B , B }
2 1 supeq1i
 |-  sup ( { B } , A , R ) = sup ( { B , B } , A , R )
3 suppr
 |-  ( ( R Or A /\ B e. A /\ B e. A ) -> sup ( { B , B } , A , R ) = if ( B R B , B , B ) )
4 3 3anidm23
 |-  ( ( R Or A /\ B e. A ) -> sup ( { B , B } , A , R ) = if ( B R B , B , B ) )
5 2 4 eqtrid
 |-  ( ( R Or A /\ B e. A ) -> sup ( { B } , A , R ) = if ( B R B , B , B ) )
6 ifid
 |-  if ( B R B , B , B ) = B
7 5 6 eqtrdi
 |-  ( ( R Or A /\ B e. A ) -> sup ( { B } , A , R ) = B )