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Theorem pwsiga 33116
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 4004 . 2 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 βŠ† 𝒫 𝑂)
2 pwidg 4621 . . 3 (𝑂 ∈ 𝑉 β†’ 𝑂 ∈ 𝒫 𝑂)
3 difss 4130 . . . . . 6 (𝑂 βˆ– π‘₯) βŠ† 𝑂
4 elpw2g 5343 . . . . . 6 (𝑂 ∈ 𝑉 β†’ ((𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂 ↔ (𝑂 βˆ– π‘₯) βŠ† 𝑂))
53, 4mpbiri 257 . . . . 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 3145 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂)
8 sspwuni 5102 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝑂 ↔ βˆͺ π‘₯ βŠ† 𝑂)
9 vuniex 7725 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
109elpw 4605 . . . . . . . 8 (βˆͺ π‘₯ ∈ 𝒫 𝑂 ↔ βˆͺ π‘₯ βŠ† 𝑂)
118, 10bitr4i 277 . . . . . . 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 4608 . . . . . 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 3145 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂))
182, 7, 173jca 1128 . 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 5375 . . 3 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ V)
20 issiga 33098 . . 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 711 1 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   ∈ wcel 2106  βˆ€wral 3061  Vcvv 3474   βˆ– cdif 3944   βŠ† wss 3947  π’« cpw 4601  βˆͺ cuni 4907   class class class wbr 5147  β€˜cfv 6540  Ο‰com 7851   β‰Ό cdom 8933  sigAlgebracsiga 33094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-siga 33095
This theorem is referenced by:  sigagenval  33126  dmsigagen  33130  ldsysgenld  33146  pwcntmeas  33213  ddemeas  33222  mbfmcnt  33255
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