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Theorem pwsiga 30321
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 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))

Proof of Theorem pwsiga
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssid 3657 . . 3 𝒫 𝑂 ⊆ 𝒫 𝑂
21a1i 11 . 2 (𝑂𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
3 pwidg 4206 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
4 difss 3770 . . . . . 6 (𝑂𝑥) ⊆ 𝑂
5 elpw2g 4857 . . . . . 6 (𝑂𝑉 → ((𝑂𝑥) ∈ 𝒫 𝑂 ↔ (𝑂𝑥) ⊆ 𝑂))
64, 5mpbiri 248 . . . . 5 (𝑂𝑉 → (𝑂𝑥) ∈ 𝒫 𝑂)
76a1d 25 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂𝑥) ∈ 𝒫 𝑂))
87ralrimiv 2994 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
9 sspwuni 4643 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
10 vuniex 6996 . . . . . . . . 9 𝑥 ∈ V
1110elpw 4197 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
129, 11bitr4i 267 . . . . . . 7 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1312biimpi 206 . . . . . 6 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1413a1d 25 . . . . 5 (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
15 elpwi 4201 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
1615imim1i 63 . . . . 5 ((𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)) → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
1714, 16mp1i 13 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
1817ralrimiv 2994 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
193, 8, 183jca 1261 . 2 (𝑂𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
20 pwexg 4880 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
21 issiga 30302 . . 3 (𝒫 𝑂 ∈ V → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))))
2220, 21syl 17 . 2 (𝑂𝑉 → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))))
232, 19, 22mpbir2and 977 1 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  cfv 5926  ωcom 7107  cdom 7995  sigAlgebracsiga 30298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-siga 30299
This theorem is referenced by:  sigagenval  30331  dmsigagen  30335  ldsysgenld  30351  pwcntmeas  30418  ddemeas  30427  mbfmcnt  30458
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