Theorem probtot 34923

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 34920	. 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 2145  ∪ cuni 4867  dom cdm 5655  ran crn 5656  ‘cfv 6533  1c1 11125  measurescmeas 34706  Probcprb 34918

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5665  df-iota 6489  df-fv 6541  df-prob 34919

This theorem is used by:  prob01  34924  probdsb  34933  dstrvprob  34983  dstfrvclim1  34989