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| Mirrors > Home > MPE Home > Th. List > Mathboxes > 0sald | Structured version Visualization version GIF version | ||
| Description: The empty set belongs to every sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) | 
| Ref | Expression | 
|---|---|
| 0sald.1 | ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| Ref | Expression | 
|---|---|
| 0sald | ⊢ (𝜑 → ∅ ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | 0sald.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 2 | 0sal 46340 | . 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 ∅c0 4332 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: subsalsal 46379 smfpimltxr 46767 smfconst 46769 smfpimgtxr 46800 smfresal 46808 | 
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