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Theorem ome0 45879
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 45876 . . . . 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 767 . 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 1534  wcel 2099  wral 3057  c0 4318  𝒫 cpw 4598   cuni 4903   class class class wbr 5142  dom cdm 5672  cres 5674  wf 6538  cfv 6542  (class class class)co 7414  ωcom 7864  cdom 8955  0cc0 11132  +∞cpnf 11269  cle 11273  [,]cicc 13353  Σ^csumge0 45744  OutMeascome 45871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-ome 45872
This theorem is referenced by:  caragen0  45888  caragenunidm  45890  caratheodory  45910
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