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Theorem pwsiga 30962
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 ssidd 3906 . 2 (𝑂𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
2 pwidg 4462 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 difss 4024 . . . . . 6 (𝑂𝑥) ⊆ 𝑂
4 elpw2g 5131 . . . . . 6 (𝑂𝑉 → ((𝑂𝑥) ∈ 𝒫 𝑂 ↔ (𝑂𝑥) ⊆ 𝑂))
53, 4mpbiri 259 . . . . 5 (𝑂𝑉 → (𝑂𝑥) ∈ 𝒫 𝑂)
65a1d 25 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂𝑥) ∈ 𝒫 𝑂))
76ralrimiv 3146 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
8 sspwuni 4915 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
9 vuniex 7315 . . . . . . . . 9 𝑥 ∈ V
109elpw 4453 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
118, 10bitr4i 279 . . . . . . 7 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1211biimpi 217 . . . . . 6 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1312a1d 25 . . . . 5 (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
14 elpwi 4457 . . . . . 6 (𝑥 ∈ 𝒫 𝒫 𝑂𝑥 ⊆ 𝒫 𝑂)
1514imim1i 63 . . . . 5 ((𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)) → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
1613, 15mp1i 13 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
1716ralrimiv 3146 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
182, 7, 173jca 1119 . 2 (𝑂𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
19 pwexg 5163 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
20 issiga 30944 . . 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)))))
2119, 20syl 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)))))
221, 18, 21mpbir2and 709 1 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1078  wcel 2079  wral 3103  Vcvv 3432  cdif 3851  wss 3854  𝒫 cpw 4447   cuni 4739   class class class wbr 4956  cfv 6217  ωcom 7427  cdom 8345  sigAlgebracsiga 30940
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-fal 1533  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-iota 6181  df-fun 6219  df-fv 6225  df-siga 30941
This theorem is referenced by:  sigagenval  30972  dmsigagen  30976  ldsysgenld  30992  pwcntmeas  31059  ddemeas  31068  mbfmcnt  31099
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