Description: A probability is an element of [ 0 , 1 ]. First axiom of Kolmogorov. (Contributed by Thierry Arnoux, 25-Dec-2016)
Ref | Expression | ||
---|---|---|---|
Assertion | prob01 | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domprobmeas | |
|
2 | measvxrge0 | |
|
3 | 1 2 | sylan | |
4 | elxrge0 | |
|
5 | 3 4 | sylib | |
6 | 1 | adantr | |
7 | simpr | |
|
8 | measbase | |
|
9 | unielsiga | |
|
10 | 6 8 9 | 3syl | |
11 | elssuni | |
|
12 | 11 | adantl | |
13 | 6 7 10 12 | measssd | |
14 | probtot | |
|
15 | 14 | breq2d | |
16 | 15 | adantr | |
17 | 13 16 | mpbid | |
18 | df-3an | |
|
19 | 5 17 18 | sylanbrc | |
20 | 0xr | |
|
21 | 1xr | |
|
22 | elicc1 | |
|
23 | 20 21 22 | mp2an | |
24 | 19 23 | sylibr | |