Theorem mea0 46439

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 46436	. . 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 218	. 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 769	1 ⊢ (𝜑 → (𝑀‘∅) = 0)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∅c0 4286  𝒫 cpw 4553  ∪ cuni 4861  Disj wdisj 5062   class class class wbr 5095  dom cdm 5623   ↾ cres 5625  ⟶wf 6482  ‘cfv 6486  (class class class)co 7353  ωcom 7806   ≼ cdom 8877  0cc0 11028  +∞cpnf 11165  [,]cicc 13269  SAlgcsalg 46293  Σ^{^}csumge0 46347  Meascmea 46434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-mea 46435

This theorem is referenced by:  meadjun  46447  meadjiunlem  46450  vonioo  46667  vonicc  46670