Theorem elsigagen2 31517

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 1133	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐴 ∈ 𝑉)
2	1	sgsiga 31511	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
3		sssigagen 31514	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
4		sspw 4510	. . . 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 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ⊆ 𝐴)
7		simp3 1135	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
8		ctex 8507	. . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
9		elpwg 4500	. . . . 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 260	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
12	5, 11	sseldd 3916	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
13		sigaclcu 31486	. 2 ⊢ (((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
14	2, 12, 7, 13	syl3anc 1368	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   ∈ wcel 2111  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ran crn 5520  ‘cfv 6324  ωcom 7560   ≼ cdom 8490  sigAlgebracsiga 31477  sigaGencsigagen 31507

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-dom 8494  df-siga 31478  df-sigagen 31508

This theorem is referenced by:  sxbrsigalem1  31653