| 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 4378 | . 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆 | |
| 2 | 0sal 46335 | . . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
| 3 | saldifcl 46334 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆) | |
| 4 | 2, 3 | mpdan 687 | . 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆) |
| 5 | 1, 4 | eqeltrrid 2846 | 1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ∖ cdif 3948 ∅c0 4333 ∪ cuni 4907 SAlgcsalg 46323 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-ss 3968 df-nul 4334 df-pw 4602 df-uni 4908 df-salg 46324 |
| This theorem is referenced by: intsaluni 46344 unisalgen 46355 salgencntex 46358 salunid 46368 |
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