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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 46334 | . . 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 1142 | 1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1086 ∈ wcel 2107 ∀wral 3060 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 ∪ cuni 4906 class class class wbr 5142 ωcom 7888 ≼ cdom 8984 SAlgcsalg 46328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-ss 3967 df-pw 4601 df-uni 4907 df-salg 46329 | 
| This theorem is referenced by: saluni 46345 intsal 46350 0sald 46370 ismeannd 46487 | 
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