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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 4329 | . 2 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆 | |
2 | 0sal 42482 | . . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
3 | saldifcl 42481 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆) | |
4 | 2, 3 | mpdan 683 | . 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆) |
5 | 1, 4 | eqeltrrid 2915 | 1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∖ cdif 3930 ∅c0 4288 ∪ cuni 4830 SAlgcsalg 42470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-in 3940 df-ss 3949 df-nul 4289 df-pw 4537 df-uni 4831 df-salg 42471 |
This theorem is referenced by: intsaluni 42489 unisalgen 42500 salgencntex 42503 salunid 42513 |
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