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Theorem pwsiga 31511
 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 3938 . 2 (𝑂𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
2 pwidg 4519 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 difss 4059 . . . . . 6 (𝑂𝑥) ⊆ 𝑂
4 elpw2g 5211 . . . . . 6 (𝑂𝑉 → ((𝑂𝑥) ∈ 𝒫 𝑂 ↔ (𝑂𝑥) ⊆ 𝑂))
53, 4mpbiri 261 . . . . 5 (𝑂𝑉 → (𝑂𝑥) ∈ 𝒫 𝑂)
65a1d 25 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂𝑥) ∈ 𝒫 𝑂))
76ralrimiv 3148 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
8 sspwuni 4985 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
9 vuniex 7447 . . . . . . . . 9 𝑥 ∈ V
109elpw 4501 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
118, 10bitr4i 281 . . . . . . 7 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1211biimpi 219 . . . . . 6 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1312a1d 25 . . . . 5 (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
14 elpwi 4506 . . . . . 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 3148 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
182, 7, 173jca 1125 . 2 (𝑂𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
19 pwexg 5244 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
20 issiga 31493 . . 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 712 1 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ‘cfv 6324  ωcom 7562   ≼ cdom 8492  sigAlgebracsiga 31489 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-siga 31490 This theorem is referenced by:  sigagenval  31521  dmsigagen  31525  ldsysgenld  31541  pwcntmeas  31608  ddemeas  31617  mbfmcnt  31648
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