Description: If a subset of S contains the GLB of S , then the two sets have the same GLB. (Contributed by Zhi Wang, 26-Sep-2024)
Ref | Expression | ||
---|---|---|---|
Hypotheses | lubsscl.k | |
|
lubsscl.t | |
||
glbsscl.g | |
||
glbsscl.s | |
||
glbsscl.x | |
||
Assertion | glbsscl | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lubsscl.k | |
|
2 | lubsscl.t | |
|
3 | glbsscl.g | |
|
4 | glbsscl.s | |
|
5 | glbsscl.x | |
|
6 | eqid | |
|
7 | eqid | |
|
8 | 6 7 3 1 4 | glbelss | |
9 | 2 8 | sstrd | |
10 | 9 5 | sseldd | |
11 | 1 | adantr | |
12 | 4 | adantr | |
13 | 2 | sselda | |
14 | 6 7 3 11 12 13 | glble | |
15 | 14 | ralrimiva | |
16 | breq2 | |
|
17 | simp3 | |
|
18 | 5 | 3ad2ant1 | |
19 | 16 17 18 | rspcdva | |
20 | 19 | 3expia | |
21 | 20 | ralrimiva | |
22 | breq1 | |
|
23 | 22 | ralbidv | |
24 | breq2 | |
|
25 | 24 | imbi2d | |
26 | 25 | ralbidv | |
27 | 23 26 | anbi12d | |
28 | 27 | rspcev | |
29 | 10 15 21 28 | syl12anc | |
30 | biid | |
|
31 | 6 7 3 30 1 | glbeldm2 | |
32 | 9 29 31 | mpbir2and | |
33 | eqidd | |
|
34 | 3 | a1i | |
35 | 7 33 34 1 9 10 14 19 | posglbdg | |
36 | 32 35 | jca | |