Theorem probtot 34743

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 34740	. 2 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simprbi 502	1 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∪ cuni 4873  dom cdm 5659  ran crn 5660  ‘cfv 6533  1c1 11097  measurescmeas 34526  Probcprb 34738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-dm 5669  df-iota 6489  df-fv 6541  df-prob 34739

This theorem is referenced by:  prob01  34744  probdsb  34753  dstrvprob  34803  dstfrvclim1  34809