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 | id 22 | . . 3 ⊢ (𝑆 ∈ SAlg → 𝑆 ∈ SAlg) | |
2 | issal 39210 | . . . 4 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))) |
4 | 1, 3 | mpbid 221 | . 2 ⊢ (𝑆 ∈ SAlg → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))) |
5 | 4 | simp1d 1066 | 1 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 ∈ wcel 1977 ∀wral 2896 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 ωcom 6957 ≼ cdom 7839 SAlgcsalg 39204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-pw 4110 df-uni 4373 df-salg 39205 |
This theorem is referenced by: saluni 39220 intsal 39224 0sald 39244 ismeannd 39360 |
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