Theorem pwsiga 34127

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 3973	. 2 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ⊆ 𝒫 𝑂)
2		pwidg 4586	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
3		difss 4102	. . . . . 6 ⊢ (𝑂 ∖ 𝑥) ⊆ 𝑂
4		elpw2g 5291	. . . . . 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 3125	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 𝑂(𝑂 ∖ 𝑥) ∈ 𝒫 𝑂)
8		sspwuni 5067	. . . . . . . 8 ⊢ (𝑥 ⊆ 𝒫 𝑂 ↔ ∪ 𝑥 ⊆ 𝑂)
9		vuniex 7718	. . . . . . . . 9 ⊢ ∪ 𝑥 ∈ V
10	9	elpw 4570	. . . . . . . 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 4573	. . . . . 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 3125	. . 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 5336	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ V)
20		issiga 34109	. . 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 3045  Vcvv 3450   ∖ cdif 3914   ⊆ wss 3917  𝒫 cpw 4566  ∪ cuni 4874   class class class wbr 5110  ‘cfv 6514  ωcom 7845   ≼ cdom 8919  sigAlgebracsiga 34105

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 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-siga 34106

This theorem is referenced by:  sigagenval  34137  dmsigagen  34141  ldsysgenld  34157  pwcntmeas  34224  ddemeas  34233  mbfmcnt  34266