Theorem probtot 35094

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 35091	. 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 1569   ∈ wcel 2144  ∪ cuni 4866  dom cdm 5651  ran crn 5652  ‘cfv 6543  1c1 11208  measurescmeas 34878  Probcprb 35089

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-dm 5661  df-iota 6497  df-fv 6551  df-prob 35090

This theorem is used by:  prob01  35095  probdsb  35104  dstrvprob  35154  dstfrvclim1  35160