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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 34391 | . 2 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) | |
2 | 1 | simprbi 496 | 1 ⊢ (𝑃 ∈ Prob → (𝑃‘∪ dom 𝑃) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 dom cdm 5689 ran crn 5690 ‘cfv 6563 1c1 11154 measurescmeas 34176 Probcprb 34389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-dm 5699 df-iota 6516 df-fv 6571 df-prob 34390 |
This theorem is referenced by: prob01 34395 probdsb 34404 dstrvprob 34453 dstfrvclim1 34459 |
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