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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 4384 | . 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆 | |
2 | 0sal 46276 | . . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
3 | saldifcl 46275 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆) | |
4 | 2, 3 | mpdan 687 | . 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆) |
5 | 1, 4 | eqeltrrid 2844 | 1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∖ cdif 3960 ∅c0 4339 ∪ cuni 4912 SAlgcsalg 46264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rab 3434 df-v 3480 df-dif 3966 df-ss 3980 df-nul 4340 df-pw 4607 df-uni 4913 df-salg 46265 |
This theorem is referenced by: intsaluni 46285 unisalgen 46296 salgencntex 46299 salunid 46309 |
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