![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 46276 | . 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∅c0 4339 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: subsalsal 46315 smfpimltxr 46703 smfconst 46705 smfpimgtxr 46736 smfresal 46744 |
Copyright terms: Public domain | W3C validator |