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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 31404 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴)) | |
2 | 1 | sselda 3967 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ (sigaGen‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ‘cfv 6355 sigaGencsigagen 31397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fv 6363 df-siga 31368 df-sigagen 31398 |
This theorem is referenced by: cldssbrsiga 31446 dya2iocbrsiga 31533 dya2icobrsiga 31534 sxbrsigalem2 31544 |
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