Metamath Proof Explorer


Theorem ovncl

Description: The Lebesgue outer measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 11-Oct-2020)

Ref Expression
Hypotheses ovncl.1 φ X Fin
ovncl.2 φ A X
Assertion ovncl φ voln* X A 0 +∞

Proof

Step Hyp Ref Expression
1 ovncl.1 φ X Fin
2 ovncl.2 φ A X
3 1 ovnf φ voln* X : 𝒫 X 0 +∞
4 ovexd φ X V
5 4 2 ssexd φ A V
6 elpwg A V A 𝒫 X A X
7 5 6 syl φ A 𝒫 X A X
8 2 7 mpbird φ A 𝒫 X
9 3 8 ffvelrnd φ voln* X A 0 +∞