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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsigass | Structured version Visualization version GIF version |
Description: An element of a sigma-algebra is a subset of the base set. (Contributed by Thierry Arnoux, 6-Jun-2017.) |
Ref | Expression |
---|---|
elsigass | ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sgon 31378 | . . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ∈ (sigAlgebra‘∪ 𝑆)) | |
2 | sigasspw 31370 | . . . 4 ⊢ (𝑆 ∈ (sigAlgebra‘∪ 𝑆) → 𝑆 ⊆ 𝒫 ∪ 𝑆) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → 𝑆 ⊆ 𝒫 ∪ 𝑆) |
4 | 3 | sselda 3967 | . 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ∈ 𝒫 ∪ 𝑆) |
5 | 4 | elpwid 4553 | 1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆) → 𝐴 ⊆ ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4832 ran crn 5551 ‘cfv 6350 sigAlgebracsiga 31362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-fv 6358 df-siga 31363 |
This theorem is referenced by: (None) |
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