Theorem unelsiga 34318

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 4854	. . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
2	1	3adant1 1136	. 2 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∪ {𝐴, 𝐵} = (𝐴 ∪ 𝐵))
3		isrnsigau 34311	. . . . . 6 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (𝑆 ⊆ 𝒫 ∪ 𝑆 ∧ (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
4	3	simprd 496	. . . . 5 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → (∪ 𝑆 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (∪ 𝑆 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))
5	4	simp3d 1150	. . . 4 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
6	5	3ad2ant1 1139	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))
7		prct 32805	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ≼ ω)
8	7	3adant1 1136	. . 3 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ≼ ω)
9		prelpwi 5386	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10		breq1 5075	. . . . . . 7 ⊢ (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11		unieq 4849	. . . . . . . 8 ⊢ (𝑥 = {𝐴, 𝐵} → ∪ 𝑥 = ∪ {𝐴, 𝐵})
12	11	eleq1d 2824	. . . . . . 7 ⊢ (𝑥 = {𝐴, 𝐵} → (∪ 𝑥 ∈ 𝑆 ↔ ∪ {𝐴, 𝐵} ∈ 𝑆))
13	10, 12	imbi12d 345	. . . . . 6 ⊢ (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆) ↔ ({𝐴, 𝐵} ≼ ω → ∪ {𝐴, 𝐵} ∈ 𝑆)))
14	13	rspcv 3556	. . . . 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 1136	. . 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 2840	1 ⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐴 ∪ 𝐵) ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4529  {cpr 4557  ∪ cuni 4838   class class class wbr 5072  ran crn 5619  ωcom 7806   ≼ cdom 8881  sigAlgebracsiga 34292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-oi 9415  df-dju 9816  df-card 9854  df-siga 34293

This theorem is referenced by:  measun  34395  aean  34428  sibfof  34524