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Mathbox for Thierry Arnoux |
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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 34354 | . 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
2 | 0elsiga 34056 | . 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 2104 ∅c0 4339 ∪ cuni 4914 dom cdm 5683 ran crn 5684 sigAlgebracsiga 34050 Probcprb 34350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-fv 6566 df-ov 7428 df-esum 33970 df-siga 34051 df-meas 34138 df-prob 34351 |
This theorem is referenced by: cndprobnul 34380 |
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