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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saluni | Structured version Visualization version GIF version |
Description: A set is an element of any sigma-algebra on it. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
saluni | ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4352 | . 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆 | |
2 | 0sal 44714 | . . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
3 | saldifcl 44713 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆) | |
4 | 2, 3 | mpdan 685 | . 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆) |
5 | 1, 4 | eqeltrrid 2837 | 1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∖ cdif 3925 ∅c0 4302 ∪ cuni 4885 SAlgcsalg 44702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rab 3419 df-v 3461 df-dif 3931 df-in 3935 df-ss 3945 df-nul 4303 df-pw 4582 df-uni 4886 df-salg 44703 |
This theorem is referenced by: intsaluni 44723 unisalgen 44734 salgencntex 44737 salunid 44747 |
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