Description: Two Hilbert lattice elements in a proper subset relationship imply the existence of an atom less than or equal to one but not the other. (Contributed by NM, 10-Jun-2004) (New usage is discouraged.)
Ref | Expression | ||
---|---|---|---|
Hypotheses | chpssat.1 | |
|
chpssat.2 | |
||
Assertion | chpssati | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | chpssat.1 | |
|
2 | chpssat.2 | |
|
3 | dfpss3 | |
|
4 | 3 | simprbi | |
5 | iman | |
|
6 | 5 | ralbii | |
7 | ss2rab | |
|
8 | ssrab2 | |
|
9 | atssch | |
|
10 | 8 9 | sstri | |
11 | ssrab2 | |
|
12 | 11 9 | sstri | |
13 | chsupss | |
|
14 | 10 12 13 | mp2an | |
15 | 2 | hatomistici | |
16 | 1 | hatomistici | |
17 | 14 15 16 | 3sstr4g | |
18 | 7 17 | sylbir | |
19 | 6 18 | sylbir | |
20 | 19 | con3i | |
21 | dfrex2 | |
|
22 | 20 21 | sylibr | |
23 | 4 22 | syl | |