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Theorem unelsiga 31501
Description: A sigma-algebra is closed under pairwise unions. (Contributed by Thierry Arnoux, 13-Dec-2016.)
Assertion
Ref Expression
unelsiga ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)

Proof of Theorem unelsiga
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 uniprg 4821 . . 3 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1127 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
3 isrnsigau 31494 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43simprd 499 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
54simp3d 1141 . . . 4 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
653ad2ant1 1130 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
7 prct 30479 . . . 4 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
873adant1 1127 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
9 prelpwi 5308 . . . . 5 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10 breq1 5036 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11 unieq 4814 . . . . . . . 8 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
1211eleq1d 2877 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → ( 𝑥𝑆 {𝐴, 𝐵} ∈ 𝑆))
1310, 12imbi12d 348 . . . . . 6 (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
1413rspcv 3569 . . . . 5 ({𝐴, 𝐵} ∈ 𝒫 𝑆 → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
159, 14syl 17 . . . 4 ((𝐴𝑆𝐵𝑆) → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
16153adant1 1127 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
176, 8, 16mp2d 49 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝑆)
182, 17eqeltrrd 2894 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  cdif 3881  cun 3882  wss 3884  𝒫 cpw 4500  {cpr 4530   cuni 4803   class class class wbr 5033  ran crn 5524  ωcom 7564  cdom 8494  sigAlgebracsiga 31475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-dju 9318  df-card 9356  df-siga 31476
This theorem is referenced by:  measun  31578  aean  31611  sibfof  31706
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