Description: If an infinite set A is included in the underlying set of a finite cover B , then there exists a set of the cover that contains an infinite number of element of A . (Contributed by FL, 2-Aug-2009)
Ref | Expression | ||
---|---|---|---|
Assertion | infssuni | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfral2 | |
|
2 | iunfi | |
|
3 | iunin2 | |
|
4 | 3 | eleq1i | |
5 | uniiun | |
|
6 | 5 | eqcomi | |
7 | 6 | ineq2i | |
8 | 7 | eleq1i | |
9 | df-ss | |
|
10 | eleq1 | |
|
11 | pm2.24 | |
|
12 | 10 11 | syl6bi | |
13 | 9 12 | sylbi | |
14 | 13 | com12 | |
15 | 8 14 | sylbi | |
16 | 4 15 | sylbi | |
17 | 2 16 | syl | |
18 | 17 | ex | |
19 | 18 | com24 | |
20 | 19 | 3imp21 | |
21 | 1 20 | biimtrrid | |
22 | 21 | pm2.18d | |