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Theorem omess0 43160
Description: If the outer measure of a set is 0, then the outer measure of its subsets is 0. (Contributed by Glauco Siliprandi, 24-Dec-2020.)
Hypotheses
Ref Expression
omess0.o (𝜑𝑂 ∈ OutMeas)
omess0.x 𝑋 = dom 𝑂
omess0.a (𝜑𝐴𝑋)
omess0.z (𝜑 → (𝑂𝐴) = 0)
omess0.s (𝜑𝐵𝐴)
Assertion
Ref Expression
omess0 (𝜑 → (𝑂𝐵) = 0)

Proof of Theorem omess0
StepHypRef Expression
1 omess0.o . . 3 (𝜑𝑂 ∈ OutMeas)
2 omess0.x . . 3 𝑋 = dom 𝑂
3 omess0.s . . . 4 (𝜑𝐵𝐴)
4 omess0.a . . . 4 (𝜑𝐴𝑋)
53, 4sstrd 3928 . . 3 (𝜑𝐵𝑋)
61, 2, 5omexrcl 43133 . 2 (𝜑 → (𝑂𝐵) ∈ ℝ*)
7 0xr 10681 . . 3 0 ∈ ℝ*
87a1i 11 . 2 (𝜑 → 0 ∈ ℝ*)
91, 2, 4, 3omessle 43124 . . 3 (𝜑 → (𝑂𝐵) ≤ (𝑂𝐴))
10 omess0.z . . 3 (𝜑 → (𝑂𝐴) = 0)
119, 10breqtrd 5059 . 2 (𝜑 → (𝑂𝐵) ≤ 0)
121, 2, 5omege0 43159 . 2 (𝜑 → 0 ≤ (𝑂𝐵))
136, 8, 11, 12xrletrid 12540 1 (𝜑 → (𝑂𝐵) = 0)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2112  wss 3884   cuni 4803  dom cdm 5523  cfv 6328  0cc0 10530  *cxr 10667  cle 10669  OutMeascome 43115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-addrcl 10591  ax-rnegex 10601  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-po 5442  df-so 5443  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-icc 12737  df-ome 43116
This theorem is referenced by:  caragencmpl  43161  voncmpl  43247
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