Description: A Hilbert lattice has a height of at least 2. (Contributed by NM, 4-Dec-2011)
Ref | Expression | ||
---|---|---|---|
Hypotheses | hlhgt4.b | |
|
hlhgt4.s | |
||
hlhgt4.z | |
||
hlhgt4.u | |
||
Assertion | hlhgt2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhgt4.b | |
|
2 | hlhgt4.s | |
|
3 | hlhgt4.z | |
|
4 | hlhgt4.u | |
|
5 | 1 2 3 4 | hlhgt4 | |
6 | hlpos | |
|
7 | 6 | ad3antrrr | |
8 | hlop | |
|
9 | 8 | ad3antrrr | |
10 | 1 3 | op0cl | |
11 | 9 10 | syl | |
12 | simpllr | |
|
13 | simplr | |
|
14 | 1 2 | plttr | |
15 | 7 11 12 13 14 | syl13anc | |
16 | simpr | |
|
17 | 1 4 | op1cl | |
18 | 9 17 | syl | |
19 | 1 2 | plttr | |
20 | 7 13 16 18 19 | syl13anc | |
21 | 15 20 | anim12d | |
22 | 21 | rexlimdva | |
23 | 22 | reximdva | |
24 | 23 | rexlimdva | |
25 | 5 24 | mpd | |