Metamath Proof Explorer


Theorem omecl

Description: The outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses omecl.o φOOutMeas
omecl.x X=domO
omecl.ss φAX
Assertion omecl φOA0+∞

Proof

Step Hyp Ref Expression
1 omecl.o φOOutMeas
2 omecl.x X=domO
3 omecl.ss φAX
4 1 2 omef φO:𝒫X0+∞
5 2 a1i φX=domO
6 1 dmexd φdomOV
7 6 uniexd φdomOV
8 5 7 eqeltrd φXV
9 8 3 ssexd φAV
10 elpwg AVA𝒫XAX
11 9 10 syl φA𝒫XAX
12 3 11 mpbird φA𝒫X
13 4 12 ffvelcdmd φOA0+∞