Mathbox for Glauco Siliprandi |
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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 42598 | . 2 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∅c0 4291 SAlgcsalg 42586 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-dif 3939 df-in 3943 df-ss 3952 df-pw 4541 df-uni 4833 df-salg 42587 |
This theorem is referenced by: subsalsal 42635 smfpimltxr 43017 smfconst 43019 smfpimgtxr 43049 smfresal 43056 |
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