Description: If the outer measure of a set is real, then the outer measure of any of its subset is real. (Contributed by Glauco Siliprandi, 17-Aug-2020)
Ref | Expression | ||
---|---|---|---|
Hypotheses | omessre.o | |
|
omessre.x | |
||
omessre.a | |
||
omessre.re | |
||
omessre.b | |
||
Assertion | omessre | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omessre.o | |
|
2 | omessre.x | |
|
3 | omessre.a | |
|
4 | omessre.re | |
|
5 | omessre.b | |
|
6 | rge0ssre | |
|
7 | 0xr | |
|
8 | 7 | a1i | |
9 | pnfxr | |
|
10 | 9 | a1i | |
11 | 5 3 | sstrd | |
12 | 1 2 11 | omexrcl | |
13 | 1 2 11 | omecl | |
14 | iccgelb | |
|
15 | 8 10 13 14 | syl3anc | |
16 | 4 | rexrd | |
17 | 1 2 3 5 | omessle | |
18 | 4 | ltpnfd | |
19 | 12 16 10 17 18 | xrlelttrd | |
20 | 8 10 12 15 19 | elicod | |
21 | 6 20 | sselid | |