Theorem elsigagen2 31411

Description: Any countable union of elements of a set is also in the sigma-algebra that set generates. (Contributed by Thierry Arnoux, 17-Sep-2017.)

Assertion

Ref	Expression
elsigagen2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))

Proof of Theorem elsigagen2

Step	Hyp	Ref	Expression
1		simp1 1132	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐴 ∈ 𝑉)
2	1	sgsiga 31405	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
3		sssigagen 31408	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
4		sspw 4555	. . . 4 ⊢ (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
5	1, 3, 4	3syl 18	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
6		simp2 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ⊆ 𝐴)
7		simp3 1134	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
8		ctex 8527	. . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
9		elpwg 4545	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
10	7, 8, 9	3syl 18	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
11	6, 10	mpbird 259	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
12	5, 11	sseldd 3971	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
13		sigaclcu 31380	. 2 ⊢ (((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
14	2, 12, 7, 13	syl3anc 1367	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   ∈ wcel 2113  Vcvv 3497   ⊆ wss 3939  𝒫 cpw 4542  ∪ cuni 4841   class class class wbr 5069  ran crn 5559  ‘cfv 6358  ωcom 7583   ≼ cdom 8510  sigAlgebracsiga 31371  sigaGencsigagen 31401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-int 4880  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-fv 6366  df-dom 8514  df-siga 31372  df-sigagen 31402

This theorem is referenced by:  sxbrsigalem1  31547