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Theorem pwsiga 30518
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 3821 . 2 (𝑂𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
2 pwidg 4366 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
3 difss 3936 . . . . . 6 (𝑂𝑥) ⊆ 𝑂
4 elpw2g 5019 . . . . . 6 (𝑂𝑉 → ((𝑂𝑥) ∈ 𝒫 𝑂 ↔ (𝑂𝑥) ⊆ 𝑂))
53, 4mpbiri 249 . . . . 5 (𝑂𝑉 → (𝑂𝑥) ∈ 𝒫 𝑂)
65a1d 25 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂𝑥) ∈ 𝒫 𝑂))
76ralrimiv 3153 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
8 sspwuni 4803 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
9 vuniex 7184 . . . . . . . . 9 𝑥 ∈ V
109elpw 4357 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
118, 10bitr4i 269 . . . . . . 7 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1211biimpi 207 . . . . . 6 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1312a1d 25 . . . . 5 (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
14 elpwi 4361 . . . . . 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 3153 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
182, 7, 173jca 1151 . 2 (𝑂𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
19 pwexg 5048 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
20 issiga 30499 . . 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 695 1 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wcel 2156  wral 3096  Vcvv 3391  cdif 3766  wss 3769  𝒫 cpw 4351   cuni 4630   class class class wbr 4844  cfv 6101  ωcom 7295  cdom 8190  sigAlgebracsiga 30495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-siga 30496
This theorem is referenced by:  sigagenval  30528  dmsigagen  30532  ldsysgenld  30548  pwcntmeas  30615  ddemeas  30624  mbfmcnt  30655
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