Description: The set of singletons is a refinement of any open covering of the discrete topology. (Contributed by Thierry Arnoux, 9-Jan-2020)
Ref | Expression | ||
---|---|---|---|
Hypothesis | dissnref.c | |
|
Assertion | dissnref | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dissnref.c | |
|
2 | simpr | |
|
3 | 1 | unisngl | |
4 | 2 3 | eqtrdi | |
5 | simplr | |
|
6 | simprr | |
|
7 | 6 | snssd | |
8 | 5 7 | eqsstrd | |
9 | simplr | |
|
10 | simp-4r | |
|
11 | 9 10 | eleqtrrd | |
12 | eluni2 | |
|
13 | 11 12 | sylib | |
14 | 8 13 | reximddv | |
15 | 1 | eqabri | |
16 | 15 | biimpi | |
17 | 16 | adantl | |
18 | 14 17 | r19.29a | |
19 | 18 | ralrimiva | |
20 | pwexg | |
|
21 | simpr | |
|
22 | snelpwi | |
|
23 | 22 | ad2antlr | |
24 | 21 23 | eqeltrd | |
25 | 24 16 | r19.29a | |
26 | 25 | ssriv | |
27 | 26 | a1i | |
28 | 20 27 | ssexd | |
29 | 28 | adantr | |
30 | eqid | |
|
31 | eqid | |
|
32 | 30 31 | isref | |
33 | 29 32 | syl | |
34 | 4 19 33 | mpbir2and | |