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Mirrors > Home > MPE Home > Th. List > Mathboxes > nuleldmp | Structured version Visualization version GIF version |
Description: The empty set is an element of the domain of the probability. (Contributed by Thierry Arnoux, 22-Jan-2017.) |
Ref | Expression |
---|---|
nuleldmp | ⊢ (𝑃 ∈ Prob → ∅ ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domprobsiga 31673 | . 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
2 | 0elsiga 31377 | . 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → ∅ ∈ dom 𝑃) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑃 ∈ Prob → ∅ ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2113 ∅c0 4294 ∪ cuni 4841 dom cdm 5558 ran crn 5559 sigAlgebracsiga 31371 Probcprb 31669 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7162 df-esum 31291 df-siga 31372 df-meas 31459 df-prob 31670 |
This theorem is referenced by: cndprobnul 31699 |
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