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Theorem pwsiga 33658
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 4000 . 2 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 βŠ† 𝒫 𝑂)
2 pwidg 4617 . . 3 (𝑂 ∈ 𝑉 β†’ 𝑂 ∈ 𝒫 𝑂)
3 difss 4126 . . . . . 6 (𝑂 βˆ– π‘₯) βŠ† 𝑂
4 elpw2g 5337 . . . . . 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 3139 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝑂(𝑂 βˆ– π‘₯) ∈ 𝒫 𝑂)
8 sspwuni 5096 . . . . . . . 8 (π‘₯ βŠ† 𝒫 𝑂 ↔ βˆͺ π‘₯ βŠ† 𝑂)
9 vuniex 7725 . . . . . . . . 9 βˆͺ π‘₯ ∈ V
109elpw 4601 . . . . . . . 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 4604 . . . . . 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 3139 . . 3 (𝑂 ∈ 𝑉 β†’ βˆ€π‘₯ ∈ 𝒫 𝒫 𝑂(π‘₯ β‰Ό Ο‰ β†’ βˆͺ π‘₯ ∈ 𝒫 𝑂))
182, 7, 173jca 1125 . 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 5369 . . 3 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ V)
20 issiga 33640 . . 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 710 1 (𝑂 ∈ 𝑉 β†’ 𝒫 𝑂 ∈ (sigAlgebraβ€˜π‘‚))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   ∈ wcel 2098  βˆ€wral 3055  Vcvv 3468   βˆ– cdif 3940   βŠ† wss 3943  π’« cpw 4597  βˆͺ cuni 4902   class class class wbr 5141  β€˜cfv 6536  Ο‰com 7851   β‰Ό cdom 8936  sigAlgebracsiga 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-siga 33637
This theorem is referenced by:  sigagenval  33668  dmsigagen  33672  ldsysgenld  33688  pwcntmeas  33755  ddemeas  33764  mbfmcnt  33797
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