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 ) |