Theorem pwsiga 34097

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	⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))

Proof of Theorem pwsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssidd 3959	. 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
2		pwidg 4571	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
3		difss 4087	. . . . . 6 ⊢ (𝑂 ∖ 𝑥) ⊆ 𝑂
4		elpw2g 5272	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ((𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ↔ (𝑂 ∖ 𝑥) ⊆ 𝑂))
5	3, 4	mpbiri 258	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
6	5	a1d 25	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝑂 → (𝑂 ∖ 𝑥) ∈ 𝒫 𝑂))
7	6	ralrimiv 3120	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
8		sspwuni 5049	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
9		vuniex 7675	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
10	9	elpw 4555	. . . . . . . 8 ⊢ (∪ 𝑥 ∈ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
11	8, 10	bitr4i 278	. . . . . . 7 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ∈ 𝒫 𝑂)
12	11	biimpi 216	. . . . . 6 ⊢ (𝑥 ⊆ 𝒫 𝑂 → ∪ 𝑥 ∈ 𝒫 𝑂)
13	12	a1d 25	. . . . 5 ⊢ (𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))
14		elpwi 4558	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝒫 𝑂 → 𝑥 ⊆ 𝒫 𝑂)
15	14	imim1i 63	. . . . 5 ⊢ ((𝑥 ⊆ 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)) → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)))
16	13, 15	mp1i 13	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑥 ∈ 𝒫 𝒫 𝑂 → (𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)))
17	16	ralrimiv 3120	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂))
18	2, 7, 17	3jca 1128	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)))
19		pwexg 5317	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
20		issiga 34079	. . 3 ⊢ (𝒫 𝑂 ∈ V → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)))))
21	19, 20	syl 17	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝒫 𝑂 ∈ (sigAlgebra‘𝑂) ↔ (𝒫 𝑂 ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂 ∧ ∀𝑥 ∈ 𝒫 𝒫 𝑂(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝒫 𝑂)))))
22	1, 18, 21	mpbir2and 713	1 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  ‘cfv 6482  ωcom 7799   ≼ cdom 8870  sigAlgebracsiga 34075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-siga 34076

This theorem is referenced by:  sigagenval  34107  dmsigagen  34111  ldsysgenld  34127  pwcntmeas  34194  ddemeas  34203  mbfmcnt  34236