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Mathbox for Thierry Arnoux |
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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 3504 | . . 3 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ∈ V) | |
2 | issiga 34068 | . . . 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 476 | . . 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 687 | . 2 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → (𝑆 ⊆ 𝒫 𝐴 ∧ (𝐴 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝐴 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))) |
5 | 4 | simpld 494 | 1 ⊢ (𝑆 ∈ (sigAlgebra‘𝐴) → 𝑆 ⊆ 𝒫 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 ∈ wcel 2103 ∀wral 3063 Vcvv 3482 ∖ cdif 3967 ⊆ wss 3970 𝒫 cpw 4622 ∪ cuni 4931 class class class wbr 5169 ‘cfv 6572 ωcom 7899 ≼ cdom 8997 sigAlgebracsiga 34064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-sbc 3799 df-csb 3916 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-iota 6524 df-fun 6574 df-fv 6580 df-siga 34065 |
This theorem is referenced by: elsigass 34081 insiga 34093 sigapisys 34111 sigaldsys 34115 brsigasspwrn 34141 1stmbfm 34217 2ndmbfm 34218 |
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