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Mirrors > Home > MPE Home > Th. List > Mathboxes > baselsiga | Structured version Visualization version GIF version |
Description: A sigma-algebra contains its base universe set. (Contributed by Thierry Arnoux, 26-Oct-2016.) |
Ref | Expression |
---|---|
baselsiga | ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3480 | . 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V) | |
2 | issiga 33882 | . . . 4 ⊢ (𝑆 ∈ V → (𝑆 ∈ (sigAlgebra‘𝐴) ↔ (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))) | |
3 | 2 | simplbda 498 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) |
4 | 3 | simp1d 1139 | . 2 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → 𝐴 ∈ 𝑆) |
5 | 1, 4 | mpancom 686 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝐴 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 ∈ wcel 2098 ∀wral 3050 Vcvv 3461 ∖ cdif 3941 ⊆ wss 3944 𝒫 cpw 4604 ∪ cuni 4909 class class class wbr 5149 ‘cfv 6549 ωcom 7871 ≼ cdom 8962 sigAlgebracsiga 33878 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-siga 33879 |
This theorem is referenced by: unielsiga 33898 sigaldsys 33929 cldssbrsiga 33957 1stmbfm 34031 2ndmbfm 34032 unveldomd 34186 probmeasb 34201 dstrvprob 34242 |
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