Description: The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006) (Revised by Mario Carneiro, 6-Sep-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | supxrleub | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supxrlub | |
|
2 | 1 | notbid | |
3 | ralnex | |
|
4 | 2 3 | bitr4di | |
5 | supxrcl | |
|
6 | xrlenlt | |
|
7 | 5 6 | sylan | |
8 | simpl | |
|
9 | 8 | sselda | |
10 | simplr | |
|
11 | 9 10 | xrlenltd | |
12 | 11 | ralbidva | |
13 | 4 7 12 | 3bitr4d | |