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Theorem ome0 45723
Description: The outer measure of the empty set is 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
ome0.1 (𝜑𝑂 ∈ OutMeas)
Assertion
Ref Expression
ome0 (𝜑 → (𝑂‘∅) = 0)

Proof of Theorem ome0
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ome0.1 . . . 4 (𝜑𝑂 ∈ OutMeas)
2 isome 45720 . . . . 5 (𝑂 ∈ OutMeas → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥))))))
31, 2syl 17 . . . 4 (𝜑 → (𝑂 ∈ OutMeas ↔ ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥))))))
41, 3mpbid 231 . . 3 (𝜑 → ((((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂𝑦 ∈ 𝒫 𝑥(𝑂𝑦) ≤ (𝑂𝑥)) ∧ ∀𝑥 ∈ 𝒫 dom 𝑂(𝑥 ≼ ω → (𝑂 𝑥) ≤ (Σ^‘(𝑂𝑥)))))
54simplld 765 . 2 (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
65simprd 495 1 (𝜑 → (𝑂‘∅) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3053  c0 4315  𝒫 cpw 4595   cuni 4900   class class class wbr 5139  dom cdm 5667  cres 5669  wf 6530  cfv 6534  (class class class)co 7402  ωcom 7849  cdom 8934  0cc0 11107  +∞cpnf 11243  cle 11247  [,]cicc 13325  Σ^csumge0 45588  OutMeascome 45715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ome 45716
This theorem is referenced by:  caragen0  45732  caragenunidm  45734  caratheodory  45754
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