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Theorem omess0 39424
 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 3578 . . 3 (𝜑𝐵𝑋)
61, 2, 5omexrcl 39397 . 2 (𝜑 → (𝑂𝐵) ∈ ℝ*)
7 0xr 9965 . . 3 0 ∈ ℝ*
87a1i 11 . 2 (𝜑 → 0 ∈ ℝ*)
91, 2, 4, 3omessle 39388 . . 3 (𝜑 → (𝑂𝐵) ≤ (𝑂𝐴))
10 omess0.z . . 3 (𝜑 → (𝑂𝐴) = 0)
119, 10breqtrd 4609 . 2 (𝜑 → (𝑂𝐵) ≤ 0)
121, 2, 5omege0 39423 . 2 (𝜑 → 0 ≤ (𝑂𝐵))
136, 8, 11, 12xrletrid 11862 1 (𝜑 → (𝑂𝐵) = 0)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372  dom cdm 5038  ‘cfv 5804  0cc0 9815  ℝ*cxr 9952   ≤ cle 9954  OutMeascome 39379 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-i2m1 9883  ax-1ne0 9884  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-icc 12053  df-ome 39380 This theorem is referenced by:  caragencmpl  39425  voncmpl  39511
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