Description: Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018)
Ref | Expression | ||
---|---|---|---|
Hypotheses | poslubmo.l | |
|
poslubmo.b | |
||
Assertion | posglbmo | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poslubmo.l | |
|
2 | poslubmo.b | |
|
3 | simprrl | |
|
4 | breq1 | |
|
5 | 4 | ralbidv | |
6 | breq1 | |
|
7 | 5 6 | imbi12d | |
8 | simprlr | |
|
9 | simplrr | |
|
10 | 7 8 9 | rspcdva | |
11 | 3 10 | mpd | |
12 | simprll | |
|
13 | breq1 | |
|
14 | 13 | ralbidv | |
15 | breq1 | |
|
16 | 14 15 | imbi12d | |
17 | simprrr | |
|
18 | simplrl | |
|
19 | 16 17 18 | rspcdva | |
20 | 12 19 | mpd | |
21 | ancom | |
|
22 | 2 1 | posasymb | |
23 | 21 22 | bitrid | |
24 | 23 | 3expb | |
25 | 24 | ad4ant13 | |
26 | 11 20 25 | mpbi2and | |
27 | 26 | ex | |
28 | 27 | ralrimivva | |
29 | breq1 | |
|
30 | 29 | ralbidv | |
31 | breq2 | |
|
32 | 31 | imbi2d | |
33 | 32 | ralbidv | |
34 | 30 33 | anbi12d | |
35 | 34 | rmo4 | |
36 | 28 35 | sylibr | |