Description: A II-finite set contains minimal elements for every nonempty chain. (Contributed by Mario Carneiro, 16-May-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | fin2i2 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr | |
|
2 | simpll | |
|
3 | ssrab2 | |
|
4 | 3 | a1i | |
5 | simprl | |
|
6 | fin23lem7 | |
|
7 | 2 1 5 6 | syl3anc | |
8 | sorpsscmpl | |
|
9 | 8 | ad2antll | |
10 | fin2i | |
|
11 | 2 4 7 9 10 | syl22anc | |
12 | sorpssuni | |
|
13 | 9 12 | syl | |
14 | 11 13 | mpbird | |
15 | psseq2 | |
|
16 | psseq2 | |
|
17 | pssdifcom2 | |
|
18 | 15 16 17 | fin23lem11 | |
19 | 1 14 18 | sylc | |
20 | sorpssint | |
|
21 | 20 | ad2antll | |
22 | 19 21 | mpbid | |