Theorem probtot 35155

Description: The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.)

Assertion

Ref	Expression
probtot	⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)

Proof of Theorem probtot

Step	Hyp	Ref	Expression
1		elprob 35152	. 2 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simprbi 503	1 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∪ cuni 4872  dom cdm 5658  ran crn 5659  ‘cfv 6566  1c1 11240  measurescmeas 34939  Probcprb 35150

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668  df-iota 6520  df-fv 6574  df-prob 35151

This theorem is used by:  prob01  35156  probdsb  35165  dstrvprob  35215  dstfrvclim1  35221