Theorem pwsal 39211

Description: The power set of a given set is a sigma-algebra (the so called discrete sigma-algebra). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
pwsal	⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ SAlg)

Proof of Theorem pwsal

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 4760	. . . 4 ⊢ ∅ ∈ 𝒫 𝑋
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∅ ∈ 𝒫 𝑋)
3		unipw 4845	. . . . . . . 8 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	3	difeq1i 3686	. . . . . . 7 ⊢ (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5	4	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦))
6		difssd 3700	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ⊆ 𝑋)
7		difexg 4735	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ V)
8		elpwg 4116	. . . . . . . 8 ⊢ ((𝑋 ∖ 𝑦) ∈ V → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
9	7, 8	syl 17	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
10	6, 9	mpbird 246	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
11	5, 10	eqeltrd 2688	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
12	11	adantr 480	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
13	12	ralrimiva 2949	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
14		elpwi 4117	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
15		uniss 4394	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝑋 → ∪ 𝑦 ⊆ ∪ 𝒫 𝑋)
16	14, 15	syl 17	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ ∪ 𝒫 𝑋)
17	16, 3	syl6sseq 3614	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ 𝑋)
18		vuniex 6852	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
19	18	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ V)
20		elpwg 4116	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ V → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
21	19, 20	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
22	17, 21	mpbird 246	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ 𝒫 𝑋)
23	22	adantl 481	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → ∪ 𝑦 ∈ 𝒫 𝑋)
24	23	a1d 25	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
25	24	ralrimiva 2949	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
26	2, 13, 25	3jca 1235	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋)))
27		pwexg 4776	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
28		issal 39210	. . 3 ⊢ (𝒫 𝑋 ∈ V → (𝒫 𝑋 ∈ SAlg ↔ (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))))
29	27, 28	syl 17	. 2 ⊢ (𝑋 ∈ 𝑉 → (𝒫 𝑋 ∈ SAlg ↔ (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))))
30	26, 29	mpbird 246	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ SAlg)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372   class class class wbr 4583  ωcom 6957   ≼ cdom 7839  SAlgcsalg 39204

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-salg 39205

This theorem is referenced by:  salgenval  39217  salgenn0  39225  salgencntex  39237  psmeasure  39364