Theorem issal 45030

Description: Express the predicate "𝑆 is a sigma-algebra." (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
issal	⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Distinct variable group:   𝑦,𝑆

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem issal

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2823	. . 3 ⊢ (𝑥 = 𝑆 → (∅ ∈ 𝑥 ↔ ∅ ∈ 𝑆))
2		id 22	. . . 4 ⊢ (𝑥 = 𝑆 → 𝑥 = 𝑆)
3		unieq 4920	. . . . . 6 ⊢ (𝑥 = 𝑆 → ∪ 𝑥 = ∪ 𝑆)
4	3	difeq1d 4122	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑥 ∖ 𝑦) = (∪ 𝑆 ∖ 𝑦))
5	4, 2	eleq12d 2828	. . . 4 ⊢ (𝑥 = 𝑆 → ((∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
6	2, 5	raleqbidv 3343	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ↔ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆))
7		pweq 4617	. . . 4 ⊢ (𝑥 = 𝑆 → 𝒫 𝑥 = 𝒫 𝑆)
8		eleq2 2823	. . . . 5 ⊢ (𝑥 = 𝑆 → (∪ 𝑦 ∈ 𝑥 ↔ ∪ 𝑦 ∈ 𝑆))
9	8	imbi2d 341	. . . 4 ⊢ (𝑥 = 𝑆 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ (𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
10	7, 9	raleqbidv 3343	. . 3 ⊢ (𝑥 = 𝑆 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥) ↔ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
11	1, 6, 10	3anbi123d 1437	. 2 ⊢ (𝑥 = 𝑆 → ((∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥)) ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
12		df-salg 45025	. 2 ⊢ SAlg = {𝑥 ∣ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (∪ 𝑥 ∖ 𝑦) ∈ 𝑥 ∧ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑥))}
13	11, 12	elab2g 3671	1 ⊢ (𝑆 ∈ 𝑉 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ∖ cdif 3946  ∅c0 4323  𝒫 cpw 4603  ∪ cuni 4909   class class class wbr 5149  ωcom 7855   ≼ cdom 8937  SAlgcsalg 45024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-3an 1090  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-salg 45025

This theorem is referenced by:  pwsal  45031  salunicl  45032  saluncl  45033  prsal  45034  saldifcl  45035  0sal  45036  intsal  45046  issald  45049  caragensal  45241