Theorem issald 47161

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 2769	. . . . 5 ⊢ ∪ 𝑆 = 𝑋
4	3	difeq1i 4070	. . . 4 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5		issald.d	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆)
6	4, 5	eqeltrid 2864	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7	6	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
8		issald.u	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)
9	8	3expia 1139	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
10	9	ralrimiva 3154	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
11		issald.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
12		issal 47142	. . 3 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
13	11, 12	syl 18	. 2 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
14	1, 7, 10, 13	mpbir3and 1361	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3076   ∖ cdif 3896  ∅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

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  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-ss 3916  df-pw 4559  df-uni 4868  df-salg 47137

This theorem is used by:  salexct  47162  issalnnd  47173