Theorem pwsal 47143

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 5320	. . . 4 ⊢ ∅ ∈ 𝒫 𝑋
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∅ ∈ 𝒫 𝑋)
3		unipw 5425	. . . . . . . 8 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	3	difeq1i 4070	. . . . . . 7 ⊢ (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5	4	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦))
6		difssd 4084	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ⊆ 𝑋)
7		difexg 5294	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ V)
8		elpwg 4560	. . . . . . . 8 ⊢ ((𝑋 ∖ 𝑦) ∈ V → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
9	7, 8	syl 18	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑋))
10	6, 9	mpbird 260	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
11	5, 10	eqeltrd 2860	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
12	11	adantr 486	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝑋) → (∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
13	12	ralrimiva 3154	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
14		elpwi 4564	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
15	14	unissd 4877	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ ∪ 𝒫 𝑋)
16	15, 3	sseqtrdi 3971	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ 𝑋)
17		vuniex 7741	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ V)
19		elpwg 4560	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ V → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
20	18, 19	syl 18	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → (∪ 𝑦 ∈ 𝒫 𝑋 ↔ ∪ 𝑦 ⊆ 𝑋))
21	16, 20	mpbird 260	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ 𝒫 𝑋)
22	21	adantl 487	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → ∪ 𝑦 ∈ 𝒫 𝑋)
23	22	a1d 26	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 𝒫 𝑋) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
24	23	ralrimiva 3154	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
25	2, 13, 24	3jca 1146	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋)))
26		pwexg 5343	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
27		issal 47142	. . 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 18	. 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 260	1 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ SAlg)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076  Vcvv 3450   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ∪ cuni 4867   class class class wbr 5103  ωcom 7862   ≼ cdom 8950  SAlgcsalg 47136

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-pw 4559  df-sn 4585  df-pr 4587  df-uni 4868  df-salg 47137

This theorem is used by:  salgenval  47149  salgenn0  47159  salgencntex  47171  psmeasure  47299