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Theorem pwsiga 32769
Description: Any power set forms a sigma-algebra. (Contributed by Thierry Arnoux, 13-Sep-2016.) (Revised by Thierry Arnoux, 24-Oct-2016.)
Assertion
Ref Expression
pwsiga (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚))

Proof of Theorem pwsiga
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 ssidd 3972 . 2 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 βŠ† 𝒫 𝑂)
2 pwidg 4585 . . 3 (𝑂 ∈ 𝑉 β†’ 𝑂 ∈ 𝒫 𝑂)
3 difss 4096 . . . . . 6 (𝑂 βˆ– π‘₯) βŠ† 𝑂
4 elpw2g 5306 . . . . . 6 (𝑂 ∈ 𝑉 β†’ ((𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂 ↔ (𝑂 βˆ– π‘₯) βŠ† 𝑂))
53, 4mpbiri 258 . . . . 5 (𝑂 ∈ 𝑉 β†’ (𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂)
65a1d 25 . . . 4 (𝑂 ∈ 𝑉 β†’ (π‘₯ ∈ 𝒫 𝑂 β†’ (𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂))
76ralrimiv 3143 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂)
8 sspwuni 5065 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝑂 ↔ βˆͺ π‘₯ βŠ† 𝑂)
9 vuniex 7681 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
109elpw 4569 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝒫 𝑂 ↔ βˆͺ π‘₯ βŠ† 𝑂)
118, 10bitr4i 278 . . . . . . 7 (π‘₯ βŠ† 𝒫 𝑂 ↔ βˆͺ π‘₯ ∈ 𝒫 𝑂)
1211biimpi 215 . . . . . 6 (π‘₯ βŠ† 𝒫 𝑂 β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)
1312a1d 25 . . . . 5 (π‘₯ βŠ† 𝒫 𝑂 β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂))
14 elpwi 4572 . . . . . 6 (π‘₯ ∈ 𝒫 𝒫 𝑂 β†’ π‘₯ βŠ† 𝒫 𝑂)
1514imim1i 63 . . . . 5 ((π‘₯ βŠ† 𝒫 𝑂 β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)) β†’ (π‘₯ ∈ 𝒫 𝒫 𝑂 β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)))
1613, 15mp1i 13 . . . 4 (𝑂 ∈ 𝑉 β†’ (π‘₯ ∈ 𝒫 𝒫 𝑂 β†’ (π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)))
1716ralrimiv 3143 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂))
182, 7, 173jca 1129 . 2 (𝑂 ∈ 𝑉 β†’ (𝑂 ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)))
19 pwexg 5338 . . 3 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ V)
20 issiga 32751 . . 3 (𝒫 𝑂 ∈ V β†’ (𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝒫 𝑂 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)))))
2119, 20syl 17 . 2 (𝑂 ∈ 𝑉 β†’ (𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚) ↔ (𝒫 𝑂 βŠ† 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂 ∧ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂)))))
221, 18, 21mpbir2and 712 1 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   ∈ wcel 2107  βˆ€wral 3065  Vcvv 3448   βˆ– cdif 3912   βŠ† wss 3915  π’« cpw 4565  βˆͺ cuni 4870   class class class wbr 5110  β€˜cfv 6501  Ο‰com 7807   β‰Ό cdom 8888  sigAlgebracsiga 32747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6453  df-fun 6503  df-fv 6509  df-siga 32748
This theorem is referenced by:  sigagenval  32779  dmsigagen  32783  ldsysgenld  32799  pwcntmeas  32866  ddemeas  32875  mbfmcnt  32908
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