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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 42611 | . 2 ⊢ ((𝑆 ∈ SAlg ∧ 𝐸 ∈ 𝑆) → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → (∪ 𝑆 ∖ 𝐸) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ∖ cdif 3935 ∪ cuni 4840 SAlgcsalg 42600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-in 3945 df-ss 3954 df-pw 4543 df-uni 4841 df-salg 42601 |
This theorem is referenced by: subsalsal 42649 salpreimagelt 42993 salpreimalegt 42995 smfresal 43070 |
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