Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwsiga Structured version   Visualization version   GIF version

Theorem pwsiga 30550
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 3780 . . 3 𝒫 𝑂 ⊆ 𝒫 𝑂
21a1i 11 . 2 (𝑂𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
3 pwidg 4322 . . 3 (𝑂𝑉𝑂 ∈ 𝒫 𝑂)
4 difss 3895 . . . . . 6 (𝑂𝑥) ⊆ 𝑂
5 elpw2g 4972 . . . . . 6 (𝑂𝑉 → ((𝑂𝑥) ∈ 𝒫 𝑂 ↔ (𝑂𝑥) ⊆ 𝑂))
64, 5mpbiri 249 . . . . 5 (𝑂𝑉 → (𝑂𝑥) ∈ 𝒫 𝑂)
76a1d 25 . . . 4 (𝑂𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂𝑥) ∈ 𝒫 𝑂))
87ralrimiv 3117 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂)
9 sspwuni 4756 . . . . . . . 8 (𝑥 ⊆ 𝒫 𝑂 𝑥𝑂)
10 vuniex 7122 . . . . . . . . 9 𝑥 ∈ V
1110elpw 4313 . . . . . . . 8 ( 𝑥 ∈ 𝒫 𝑂 𝑥𝑂)
129, 11bitr4i 268 . . . . . . 7 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1312biimpi 207 . . . . . 6 (𝑥 ⊆ 𝒫 𝑂 𝑥 ∈ 𝒫 𝑂)
1413a1d 25 . . . . 5 (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
15 elpwi 4317 . . . . . 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 3117 . . 3 (𝑂𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂))
193, 8, 183jca 1149 . 2 (𝑂𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → 𝑥 ∈ 𝒫 𝑂)))
20 pwexg 4994 . . 3 (𝑂𝑉 → 𝒫 𝑂 ∈ V)
21 issiga 30531 . . 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 693 1 (𝑂𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 383  w3a 1098  wcel 2148  wral 3064  Vcvv 3355  cdif 3726  wss 3729  𝒫 cpw 4307   cuni 4585   class class class wbr 4797  cfv 6042  ωcom 7233  cdom 8128  sigAlgebracsiga 30527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-fal 1640  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-iota 6005  df-fun 6044  df-fv 6050  df-siga 30528
This theorem is referenced by:  sigagenval  30560  dmsigagen  30564  ldsysgenld  30580  pwcntmeas  30647  ddemeas  30656  mbfmcnt  30687
  Copyright terms: Public domain W3C validator