Theorem elsigagen2 34547

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 1153	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐴 ∈ 𝑉)
2	1	sgsiga 34541	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
3		sssigagen 34544	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
4		sspw 4572	. . . 4 ⊢ (𝐴 ⊆ (sigaGen‘𝐴) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
5	1, 3, 4	3syl 19	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝒫 𝐴 ⊆ 𝒫 (sigaGen‘𝐴))
6		simp2 1154	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ⊆ 𝐴)
7		simp3 1155	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
8		ctex 8958	. . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
9		elpwg 4564	. . . . 5 ⊢ (𝐵 ∈ V → (𝐵 ∈ 𝒫 𝐴 ↔ 𝐵 ⊆ 𝐴))
10	7, 8, 9	3syl 19	. . . 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 3937	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
13		sigaclcu 34516	. 2 ⊢ (((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
14	2, 12, 7, 13	syl3anc 1397	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1102   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  𝒫 cpw 4561  ∪ cuni 4871   class class class wbr 5108  ran crn 5661  ‘cfv 6536  ωcom 7860   ≼ cdom 8939  sigAlgebracsiga 34507  sigaGencsigagen 34537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-dom 8943  df-siga 34508  df-sigagen 34538

This theorem is used by:  sxbrsigalem1  34684