Theorem elsigagen2 32016

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 1134	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐴 ∈ 𝑉)
2	1	sgsiga 32010	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → (sigaGen‘𝐴) ∈ ∪ ran sigAlgebra)
3		sssigagen 32013	. . . 4 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ⊆ (sigaGen‘𝐴))
4		sspw 4543	. . . 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 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ⊆ 𝐴)
7		simp3 1136	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ≼ ω)
8		ctex 8708	. . . . 5 ⊢ (𝐵 ≼ ω → 𝐵 ∈ V)
9		elpwg 4533	. . . . 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 256	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 𝐴)
12	5, 11	sseldd 3918	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → 𝐵 ∈ 𝒫 (sigaGen‘𝐴))
13		sigaclcu 31985	. 2 ⊢ (((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ 𝐵 ∈ 𝒫 (sigaGen‘𝐴) ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))
14	2, 12, 7, 13	syl3anc 1369	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≼ ω) → ∪ 𝐵 ∈ (sigaGen‘𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1085   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ran crn 5581  ‘cfv 6418  ωcom 7687   ≼ cdom 8689  sigAlgebracsiga 31976  sigaGencsigagen 32006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-dom 8693  df-siga 31977  df-sigagen 32007

This theorem is referenced by:  sxbrsigalem1  32152