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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elsigagen | Structured version Visualization version GIF version |
Description: Any element of a set is also an element of the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 27-Mar-2017.) |
Ref | Expression |
---|---|
elsigagen | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssigagen 33978 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴)) | |
2 | 1 | sselda 3979 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2099 ‘cfv 6554 sigaGencsigagen 33971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-siga 33942 df-sigagen 33972 |
This theorem is referenced by: cldssbrsiga 34020 dya2iocbrsiga 34109 dya2icobrsiga 34110 sxbrsigalem2 34120 |
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