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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 45581 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
4 | 1, 2, 3 | syl2anc 583 | 1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ∖ cdif 3938 ∪ cuni 4900 SAlgcsalg 45570 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1086 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-in 3948 df-ss 3958 df-pw 4597 df-uni 4901 df-salg 45571 |
This theorem is referenced by: subsalsal 45621 salpreimagelt 45969 salpreimalegt 45971 smfresal 46050 |
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