Theorem issald 42973

Description: Sufficient condition to prove that 𝑆 is sigma-algebra. (Contributed by Glauco Siliprandi, 3-Jan-2021.)

Hypotheses

Ref	Expression
issald.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
issald.z	⊢ (𝜑 → ∅ ∈ 𝑆)
issald.x	⊢ 𝑋 = ∪ 𝑆
issald.d	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆)
issald.u	⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)

Assertion

Ref	Expression
issald	⊢ (𝜑 → 𝑆 ∈ SAlg)

Distinct variable groups:   𝑦,𝑆   𝜑,𝑦

Allowed substitution hints:   𝑉(𝑦)   𝑋(𝑦)

Proof of Theorem issald

Step	Hyp	Ref	Expression
1		issald.z	. 2 ⊢ (𝜑 → ∅ ∈ 𝑆)
2		issald.x	. . . . . 6 ⊢ 𝑋 = ∪ 𝑆
3	2	eqcomi 2807	. . . . 5 ⊢ ∪ 𝑆 = 𝑋
4	3	difeq1i 4046	. . . 4 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5		issald.d	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆)
6	4, 5	eqeltrid 2894	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7	6	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
8		issald.u	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)
9	8	3expia 1118	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
10	9	ralrimiva 3149	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
11		issald.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
12		issal 42956	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
13	11, 12	syl 17	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
14	1, 7, 10, 13	mpbir3and 1339	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ∖ cdif 3878  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ωcom 7560   ≼ cdom 8490  SAlgcsalg 42950

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rab 3115  df-v 3443  df-dif 3884  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-salg 42951

This theorem is referenced by:  salexct  42974  issalnnd  42985