Description: If U belongs to A and U is an upper bound, then U is the sup of A. (Contributed by Glauco Siliprandi, 20-Apr-2017)
Ref | Expression | ||
---|---|---|---|
Assertion | ubelsupr | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 | |
|
2 | simp2 | |
|
3 | 2 | ne0d | |
4 | 1 2 | sseldd | |
5 | simp3 | |
|
6 | brralrspcev | |
|
7 | 4 5 6 | syl2anc | |
8 | 1 3 7 | 3jca | |
9 | suprub | |
|
10 | 8 2 9 | syl2anc | |
11 | suprleub | |
|
12 | 8 4 11 | syl2anc | |
13 | 5 12 | mpbird | |
14 | suprcl | |
|
15 | 8 14 | syl | |
16 | 4 15 | letri3d | |
17 | 10 13 16 | mpbir2and | |