![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sigasspw | Structured version Visualization version GIF version |
Description: A sigma-algebra is a set of subset of the base set. (Contributed by Thierry Arnoux, 18-Jan-2017.) |
Ref | Expression |
---|---|
sigasspw | ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3413 | . . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V) | |
2 | issiga 30772 | . . . 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 | biimpa 470 | . . 3 ⊢ ((𝑆 ∈ V ∧ 𝑆 ∈ (sigAlgebra‘𝐴)) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) |
4 | 1, 3 | mpancom 678 | . 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) |
5 | 4 | simpld 490 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 ∈ wcel 2106 ∀wral 3089 Vcvv 3397 ∖ cdif 3788 ⊆ wss 3791 𝒫 cpw 4378 ∪ cuni 4671 class class class wbr 4886 ‘cfv 6135 ωcom 7343 ≼ cdom 8239 sigAlgebracsiga 30768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ral 3094 df-rex 3095 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-siga 30769 |
This theorem is referenced by: elsigass 30786 insiga 30798 sigapisys 30816 sigaldsys 30820 brsigasspwrn 30846 1stmbfm 30920 2ndmbfm 30921 |
Copyright terms: Public domain | W3C validator |