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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 41331 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
4 | 1, 2, 3 | syl2anc 581 | 1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 ∖ cdif 3796 ∪ cuni 4659 SAlgcsalg 41320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2804 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ral 3123 df-rex 3124 df-rab 3127 df-v 3417 df-dif 3802 df-in 3806 df-ss 3813 df-pw 4381 df-uni 4660 df-salg 41321 |
This theorem is referenced by: subsalsal 41369 salpreimagelt 41713 salpreimalegt 41715 smfresal 41790 |
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