Theorem salunicl 40536

Description: SAlg sigma-algebra is closed under countable union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
salunicl.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
salunicl.t	⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)
salunicl.tct	⊢ (𝜑 → 𝑇 ≼ ω)

Assertion

Ref	Expression
salunicl	⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)

Proof of Theorem salunicl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		salunicl.tct	. 2 ⊢ (𝜑 → 𝑇 ≼ ω)
2		salunicl.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝑆)
3		salunicl.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg)
4		issal 40534	. . . . . 6 ⊢ (𝑆 ∈ SAlg → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
5	3, 4	syl 17	. . . . 5 ⊢ (𝜑 → (𝑆 ∈ SAlg ↔ (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))))
6	3, 5	mpbid 222	. . . 4 ⊢ (𝜑 → (∅ ∈ 𝑆 ∧ ∀𝑦 ∈ 𝑆 (∪ 𝑆 ∖ 𝑦) ∈ 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)))
7	6	simp3d 1075	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆))
8		breq1 4656	. . . . 5 ⊢ (𝑦 = 𝑇 → (𝑦 ≼ ω ↔ 𝑇 ≼ ω))
9		unieq 4444	. . . . . 6 ⊢ (𝑦 = 𝑇 → ∪ 𝑦 = ∪ 𝑇)
10	9	eleq1d 2686	. . . . 5 ⊢ (𝑦 = 𝑇 → (∪ 𝑦 ∈ 𝑆 ↔ ∪ 𝑇 ∈ 𝑆))
11	8, 10	imbi12d 334	. . . 4 ⊢ (𝑦 = 𝑇 → ((𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆) ↔ (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆)))
12	11	rspcva 3307	. . 3 ⊢ ((𝑇 ∈ 𝒫 𝑆 ∧ ∀𝑦 ∈ 𝒫 𝑆(𝑦 ≼ ω → ∪ 𝑦 ∈ 𝑆)) → (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))
13	2, 7, 12	syl2anc 693	. 2 ⊢ (𝜑 → (𝑇 ≼ ω → ∪ 𝑇 ∈ 𝑆))
14	1, 13	mpd 15	1 ⊢ (𝜑 → ∪ 𝑇 ∈ 𝑆)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912   ∖ cdif 3571  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436   class class class wbr 4653  ωcom 7065   ≼ cdom 7953  SAlgcsalg 40528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-salg 40529

This theorem is referenced by:  saliuncl  40542  intsal  40548  smfpimbor1lem1  41005