Theorem mea0 41188

Description: The measure of the empty set is always 0 . (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
mea0.1	⊢ (𝜑 → 𝑀 ∈ Meas)

Assertion

Ref	Expression
mea0	⊢ (𝜑 → (𝑀‘∅) = 0)

Proof of Theorem mea0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mea0.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ Meas)
2		ismea 41185	. . 3 ⊢ (𝑀 ∈ Meas ↔ (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
3	1, 2	sylib 208	. 2 ⊢ (𝜑 → (((𝑀:dom 𝑀⟶(0[,]+∞) ∧ dom 𝑀 ∈ SAlg) ∧ (𝑀‘∅) = 0) ∧ ∀𝑥 ∈ 𝒫 dom 𝑀((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → (𝑀‘∪ 𝑥) = (Σ^‘(𝑀 ↾ 𝑥)))))
4	3	simplrd 753	1 ⊢ (𝜑 → (𝑀‘∅) = 0)

Syntax hints:   → wi 4   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∅c0 4063  𝒫 cpw 4297  ∪ cuni 4574  Disj wdisj 4754   class class class wbr 4786  dom cdm 5249   ↾ cres 5251  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  ωcom 7212   ≼ cdom 8107  0cc0 10138  +∞cpnf 10273  [,]cicc 12383  SAlgcsalg 41045  Σ^{^}csumge0 41096  Meascmea 41183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pr 5034

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-mea 41184

This theorem is referenced by:  meadjun  41196  meadjiunlem  41199  vonioo  41416  vonicc  41419