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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0sal | Structured version Visualization version GIF version |
Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
0sal | ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issal 46270 | . . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
2 | 1 | ibi 267 | . 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
3 | 2 | simp1d 1141 | 1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2106 ∀wral 3059 ∖ cdif 3960 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 class class class wbr 5148 ωcom 7887 ≼ cdom 8982 SAlgcsalg 46264 |
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-3an 1088 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-ss 3980 df-pw 4607 df-uni 4913 df-salg 46265 |
This theorem is referenced by: saluni 46281 intsal 46286 0sald 46306 ismeannd 46423 |
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