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Mirrors > Home > MPE Home > Th. List > Mathboxes > probtot | Structured version Visualization version GIF version |
Description: The probability of the universe set is 1. Second axiom of Kolmogorov. (Contributed by Thierry Arnoux, 8-Dec-2016.) |
Ref | Expression |
---|---|
probtot | ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1) |
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) |
Colors of variables: wff setvar class |
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 |
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