Theorem sigaclcu 31378

Description: A sigma-algebra is closed under countable union. (Contributed by Thierry Arnoux, 26-Dec-2016.)

Assertion

Ref	Expression
sigaclcu	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝐴 ∈ 𝑆)

Proof of Theorem sigaclcu

Dummy variables 𝑜 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1133	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → 𝐴 ∈ 𝒫 𝑆)
2		isrnsiga 31374	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
3	2	simprbi 499	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
4		simpr3 1192	. . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
5	4	exlimiv 1931	. . . 4 ⊢ (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
6	3, 5	syl 17	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
7	6	3ad2ant1 1129	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
8		simp3 1134	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → 𝐴 ≼ ω)
9		breq1 5071	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 ≼ ω ↔ 𝐴 ≼ ω))
10		unieq 4851	. . . . 5 ⊢ (𝑥 = 𝐴 → ∪ 𝑥 = ∪ 𝐴)
11	10	eleq1d 2899	. . . 4 ⊢ (𝑥 = 𝐴 → (∪ 𝑥 ∈ 𝑆 ↔ ∪ 𝐴 ∈ 𝑆))
12	9, 11	imbi12d 347	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) ↔ (𝐴 ≼ ω → ∪ 𝐴 ∈ 𝑆)))
13	12	rspcv 3620	. 2 ⊢ (𝐴 ∈ 𝒫 𝑆 → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → (𝐴 ≼ ω → ∪ 𝐴 ∈ 𝑆)))
14	1, 7, 8, 13	syl3c 66	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝒫 𝑆 ∧ 𝐴 ≼ ω) → ∪ 𝐴 ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  Vcvv 3496   ∖ cdif 3935   ⊆ wss 3938  𝒫 cpw 4541  ∪ cuni 4840   class class class wbr 5068  ran crn 5558  ωcom 7582   ≼ cdom 8509  sigAlgebracsiga 31369

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365  df-siga 31370

This theorem is referenced by:  sigaclcuni  31379  sigaclfu  31380  sigaclcu2  31381  sigainb  31397  elsigagen2  31409  sigaldsys  31420  measinb  31482  measres  31483  measdivcst  31485  measdivcstALTV  31486  imambfm  31522  totprobd  31686  dstrvprob  31731