Theorem issald 41342

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 2834	. . . . 5 ⊢ ∪ 𝑆 = 𝑋
4	3	difeq1i 3951	. . . 4 ⊢ (∪ 𝑆 ∖ 𝑦) = (𝑋 ∖ 𝑦)
5		issald.d	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (𝑋 ∖ 𝑦) ∈ 𝑆)
6	4, 5	syl5eqel 2910	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
7	6	ralrimiva 3175	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆)
8		issald.u	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆 ∧ 𝑦 ≼ ω) → ∪ 𝑦 ∈ 𝑆)
9	8	3expia 1156	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝒫 𝑆) → (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
10	9	ralrimiva 3175	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
11		issald.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
12		issal 41325	. . 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 1448	1 ⊢ (𝜑 → 𝑆 ∈ SAlg)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∧ w3a 1113   = wceq 1658   ∈ wcel 2166  ∀wral 3117   ∖ cdif 3795  ∅c0 4144  𝒫 cpw 4378  ∪ cuni 4658   class class class wbr 4873  ωcom 7326   ≼ cdom 8220  SAlgcsalg 41319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-in 3805  df-ss 3812  df-pw 4380  df-uni 4659  df-salg 41320

This theorem is referenced by:  salexct  41343  issalnnd  41354