Theorem probtot 31941

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 31938	. 2 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1))
2	1	simprbi 500	1 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113  ∪ cuni 4793  dom cdm 5519  ran crn 5520  ‘cfv 6333  1c1 10609  measurescmeas 31725  Probcprb 31936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-rab 3062  df-v 3399  df-un 3846  df-in 3848  df-ss 3858  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-dm 5529  df-iota 6291  df-fv 6341  df-prob 31937

This theorem is referenced by:  prob01  31942  probdsb  31951  dstrvprob  32000  dstfrvclim1  32006