Theorem pwsal 44709

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 5331	. . . 4 ⊢ ∅ ∈ 𝒫 𝑋
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∅ ∈ 𝒫 𝑋)
3		unipw 5427	. . . . . . . 8 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	3	difeq1i 4098	. . . . . . 7 ⊢ (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5	4	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦))
6		difssd 4112	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ⊆ 𝑋)
7		difexg 5304	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ V)
8		elpwg 4583	. . . . . . . 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 256	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
11	5, 10	eqeltrd 2832	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
12	11	adantr 481	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
13	12	ralrimiva 3145	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
14		elpwi 4587	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
15	14	unissd 4895	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ ∪ 𝒫 𝑋)
16	15, 3	sseqtrdi 4012	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ 𝑋)
17		vuniex 7696	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ V)
19		elpwg 4583	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ V → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
20	18, 19	syl 17	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
21	16, 20	mpbird 256	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ 𝒫 𝑋)
22	21	adantl 482	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → ∪ 𝑦 ∈ 𝒫 𝑋)
23	22	a1d 25	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
24	23	ralrimiva 3145	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
25	2, 13, 24	3jca 1128	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋)))
26		pwexg 5353	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
27		issal 44708	. . 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))))
28	26, 27	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))))
29	25, 28	mpbird 256	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ SAlg)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  Vcvv 3459   ∖ cdif 3925   ⊆ wss 3928  ∅c0 4302  𝒫 cpw 4580  ∪ cuni 4885   class class class wbr 5125  ωcom 7822   ≼ cdom 8903  SAlgcsalg 44702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pow 5340  ax-pr 5404  ax-un 7692

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-pw 4582  df-sn 4607  df-pr 4609  df-uni 4886  df-salg 44703

This theorem is referenced by:  salgenval  44715  salgenn0  44725  salgencntex  44737  psmeasure  44865