Description: Subset law for least upper bounds. ( chsupss analog.) (Contributed by NM, 20-Oct-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lublem.b | |
|
lublem.l | |
||
lublem.u | |
||
Assertion | lubss | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lublem.b | |
|
2 | lublem.l | |
|
3 | lublem.u | |
|
4 | simp1 | |
|
5 | sstr2 | |
|
6 | 5 | impcom | |
7 | 6 | 3adant1 | |
8 | 1 3 | clatlubcl | |
9 | 8 | 3adant3 | |
10 | 4 7 9 | 3jca | |
11 | simpl1 | |
|
12 | simpl2 | |
|
13 | ssel2 | |
|
14 | 13 | 3ad2antl3 | |
15 | 1 2 3 | lubub | |
16 | 11 12 14 15 | syl3anc | |
17 | 16 | ralrimiva | |
18 | 1 2 3 | lubl | |
19 | 10 17 18 | sylc | |