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Theorem ome0 41505
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 41502 . . . . 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 224 . . 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 786 . 2 (𝜑 → ((𝑂:dom 𝑂⟶(0[,]+∞) ∧ dom 𝑂 = 𝒫 dom 𝑂) ∧ (𝑂‘∅) = 0))
65simprd 491 1 (𝜑 → (𝑂‘∅) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  wral 3117  c0 4144  𝒫 cpw 4378   cuni 4658   class class class wbr 4873  dom cdm 5342  cres 5344  wf 6119  cfv 6123  (class class class)co 6905  ωcom 7326  cdom 8220  0cc0 10252  +∞cpnf 10388  cle 10392  [,]cicc 12466  Σ^csumge0 41370  OutMeascome 41497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ome 41498
This theorem is referenced by:  caragen0  41514  caragenunidm  41516  caratheodory  41536
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