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Theorem meacl 45719
Description: The measure of a set is a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
meacl.1 (𝜑𝑀 ∈ Meas)
meacl.2 𝑆 = dom 𝑀
meacl.3 (𝜑𝐴𝑆)
Assertion
Ref Expression
meacl (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))

Proof of Theorem meacl
StepHypRef Expression
1 meacl.1 . . 3 (𝜑𝑀 ∈ Meas)
2 meacl.2 . . 3 𝑆 = dom 𝑀
31, 2meaf 45714 . 2 (𝜑𝑀:𝑆⟶(0[,]+∞))
4 meacl.3 . 2 (𝜑𝐴𝑆)
53, 4ffvelcdmd 7078 1 (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  dom cdm 5667  cfv 6534  (class class class)co 7402  0cc0 11107  +∞cpnf 11244  [,]cicc 13328  Meascmea 45710
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-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pr 5418
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-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  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-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-mea 45711
This theorem is referenced by:  meaxrcl  45722  meassle  45724  meaiunlelem  45729  meage0  45736  voncl  45927
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