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Theorem meacl 46413
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 46408 . 2 (𝜑𝑀:𝑆⟶(0[,]+∞))
4 meacl.3 . 2 (𝜑𝐴𝑆)
53, 4ffvelcdmd 7104 1 (𝜑 → (𝑀𝐴) ∈ (0[,]+∞))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  dom cdm 5688  cfv 6562  (class class class)co 7430  0cc0 11152  +∞cpnf 11289  [,]cicc 13386  Meascmea 46404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-mea 46405
This theorem is referenced by:  meaxrcl  46416  meassle  46418  meaiunlelem  46423  meage0  46430  voncl  46621
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