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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saldifcld | Structured version Visualization version GIF version |
Description: The complement of an element of a sigma-algebra is in the sigma-algebra. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
saldifcld.1 | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saldifcld.2 | ⊢ (𝜑 → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saldifcld | ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saldifcld.1 | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
2 | saldifcld.2 | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑆) | |
3 | saldifcl 44650 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
4 | 1, 2, 3 | syl2anc 585 | 1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∖ cdif 3911 ∪ cuni 4869 SAlgcsalg 44639 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 398 df-3an 1090 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rab 3407 df-v 3449 df-dif 3917 df-in 3921 df-ss 3931 df-pw 4566 df-uni 4870 df-salg 44640 |
This theorem is referenced by: subsalsal 44690 salpreimagelt 45038 salpreimalegt 45040 smfresal 45119 |
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