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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 3904 | . . . 4 ⊢ (∪ 𝑆 ∖ ∅) = ∪ 𝑆 | |
2 | 1 | eqcomi 2619 | . . 3 ⊢ ∪ 𝑆 = (∪ 𝑆 ∖ ∅) |
3 | 2 | a1i 11 | . 2 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 = (∪ 𝑆 ∖ ∅)) |
4 | id 22 | . . 3 ⊢ (𝑆 ∈ SAlg → 𝑆 ∈ SAlg) | |
5 | 0sal 39216 | . . 3 ⊢ (𝑆 ∈ SAlg → ∅ ∈ 𝑆) | |
6 | saldifcl 39215 | . . 3 ⊢ ((𝑆 ∈ SAlg ∧ ∅ ∈ 𝑆) → (∪ 𝑆 ∖ ∅) ∈ 𝑆) | |
7 | 4, 5, 6 | syl2anc 691 | . 2 ⊢ (𝑆 ∈ SAlg → (∪ 𝑆 ∖ ∅) ∈ 𝑆) |
8 | 3, 7 | eqeltrd 2688 | 1 ⊢ (𝑆 ∈ SAlg → ∪ 𝑆 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 ∪ cuni 4372 SAlgcsalg 39204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-pw 4110 df-uni 4373 df-salg 39205 |
This theorem is referenced by: intsaluni 39223 unisalgen 39234 salgencntex 39237 salunid 39247 |
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