Theorem pwsal 43746

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 5273	. . . 4 ⊢ ∅ ∈ 𝒫 𝑋
2	1	a1i 11	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∅ ∈ 𝒫 𝑋)
3		unipw 5360	. . . . . . . 8 ⊢ ∪ 𝒫 𝑋 = 𝑋
4	3	difeq1i 4049	. . . . . . 7 ⊢ (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5	4	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → (∪ 𝒫 𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑦))
6		difssd 4063	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ⊆ 𝑋)
7		difexg 5246	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑋 ∖ 𝑦) ∈ V)
8		elpwg 4533	. . . . . . . 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 2839	. . . . 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 3107	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋)
14		elpwi 4539	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → 𝑦 ⊆ 𝒫 𝑋)
15	14	unissd 4846	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ ∪ 𝒫 𝑋)
16	15, 3	sseqtrdi 3967	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ⊆ 𝑋)
17		vuniex 7570	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
18	17	a1i 11	. . . . . . . 8 ⊢ (𝑦 ∈ 𝒫 𝒫 𝑋 → ∪ 𝑦 ∈ V)
19		elpwg 4533	. . . . . . . 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 481	. . . . 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 3107	. . 3 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋))
25	2, 13, 24	3jca 1126	. 2 ⊢ (𝑋 ∈ 𝑉 → (∅ ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝑋(∪ 𝒫 𝑋 ∖ 𝑦) ∈ 𝒫 𝑋 ∧ ∀𝑦 ∈ 𝒫 𝒫 𝑋(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝒫 𝑋)))
26		pwexg 5296	. . 3 ⊢ (𝑋 ∈ 𝑉 → 𝒫 𝑋 ∈ V)
27		issal 43745	. . 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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ωcom 7687   ≼ cdom 8689  SAlgcsalg 43739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-pw 4532  df-sn 4559  df-pr 4561  df-uni 4837  df-salg 43740

This theorem is referenced by:  salgenval  43752  salgenn0  43760  salgencntex  43772  psmeasure  43899