Step |
Hyp |
Ref |
Expression |
1 |
|
supxrre1 |
|- ( ( A C_ RR /\ A =/= (/) ) -> ( sup ( A , RR* , < ) e. RR <-> sup ( A , RR* , < ) < +oo ) ) |
2 |
|
id |
|- ( A C_ RR -> A C_ RR ) |
3 |
|
rexr |
|- ( x e. RR -> x e. RR* ) |
4 |
3
|
ssriv |
|- RR C_ RR* |
5 |
4
|
a1i |
|- ( A C_ RR -> RR C_ RR* ) |
6 |
2 5
|
sstrd |
|- ( A C_ RR -> A C_ RR* ) |
7 |
|
supxrbnd2 |
|- ( A C_ RR* -> ( E. x e. RR A. y e. A y <_ x <-> sup ( A , RR* , < ) < +oo ) ) |
8 |
6 7
|
syl |
|- ( A C_ RR -> ( E. x e. RR A. y e. A y <_ x <-> sup ( A , RR* , < ) < +oo ) ) |
9 |
8
|
bicomd |
|- ( A C_ RR -> ( sup ( A , RR* , < ) < +oo <-> E. x e. RR A. y e. A y <_ x ) ) |
10 |
9
|
adantr |
|- ( ( A C_ RR /\ A =/= (/) ) -> ( sup ( A , RR* , < ) < +oo <-> E. x e. RR A. y e. A y <_ x ) ) |
11 |
1 10
|
bitrd |
|- ( ( A C_ RR /\ A =/= (/) ) -> ( sup ( A , RR* , < ) e. RR <-> E. x e. RR A. y e. A y <_ x ) ) |