Theorem unelsiga 31768

Description: A sigma-algebra is closed under pairwise unions. (Contributed by Thierry Arnoux, 13-Dec-2016.)

Assertion

Ref	Expression
unelsiga	⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)

Proof of Theorem unelsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniprg 4822	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1132	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		isrnsigau 31761	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
4	3	simprd 499	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
5	4	simp3d 1146	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
6	5	3ad2ant1 1135	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
7		prct 30723	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ≼ ω)
8	7	3adant1 1132	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ≼ ω)
9		prelpwi 5317	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10		breq1 5042	. . . . . . 7 ⊢ (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11		unieq 4816	. . . . . . . 8 ⊢ (𝑥 = {𝐴, 𝐵} → ∪ 𝑥 = ∪ {𝐴, 𝐵})
12	11	eleq1d 2815	. . . . . . 7 ⊢ (𝑥 = {𝐴, 𝐵} → (∪ 𝑥 ∈ 𝑆 ↔ ∪ {𝐴, 𝐵} ∈ 𝑆))
13	10, 12	imbi12d 348	. . . . . 6 ⊢ (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) ↔ ({𝐴, 𝐵} ≼ ω → ∪ {𝐴, 𝐵} ∈ 𝑆)))
14	13	rspcv 3522	. . . . 5 ⊢ ({𝐴, 𝐵} ∈ 𝒫 𝑆 → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → ({𝐴, 𝐵} ≼ ω → ∪ {𝐴, 𝐵} ∈ 𝑆)))
15	9, 14	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → ({𝐴, 𝐵} ≼ ω → ∪ {𝐴, 𝐵} ∈ 𝑆)))
16	15	3adant1 1132	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) → ({𝐴, 𝐵} ≼ ω → ∪ {𝐴, 𝐵} ∈ 𝑆)))
17	6, 8, 16	mp2d 49	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∪ {𝐴, 𝐵} ∈ 𝑆)
18	2, 17	eqeltrrd 2832	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1089   = wceq 1543   ∈ wcel 2112  ∀wral 3051   ∖ cdif 3850   ∪ cun 3851   ⊆ wss 3853  𝒫 cpw 4499  {cpr 4529  ∪ cuni 4805   class class class wbr 5039  ran crn 5537  ωcom 7622   ≼ cdom 8602  sigAlgebracsiga 31742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501  ax-inf2 9234

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7148  df-om 7623  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-recs 8086  df-rdg 8124  df-1o 8180  df-2o 8181  df-er 8369  df-en 8605  df-dom 8606  df-sdom 8607  df-fin 8608  df-oi 9104  df-dju 9482  df-card 9520  df-siga 31743

This theorem is referenced by:  measun  31845  aean  31878  sibfof  31973