Description: The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020)
Ref | Expression | ||
---|---|---|---|
Assertion | sup0 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sup0riota | |
|
2 | 1 | 3ad2ant1 | |
3 | simp2r | |
|
4 | simpl | |
|
5 | 4 | anim1i | |
6 | 5 | 3adant1 | |
7 | breq2 | |
|
8 | 7 | notbid | |
9 | 8 | ralbidv | |
10 | 9 | riota2 | |
11 | 6 10 | syl | |
12 | 3 11 | mpbid | |
13 | 2 12 | eqtrd | |