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Theorem unelsiga 33662
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 4918 . . 3 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1127 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
3 isrnsigau 33655 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43simprd 495 . . . . 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 32446 . . . 4 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
873adant1 1127 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
9 prelpwi 5440 . . . . 5 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10 breq1 5144 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11 unieq 4913 . . . . . . . 8 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
1211eleq1d 2812 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → ( 𝑥𝑆 {𝐴, 𝐵} ∈ 𝑆))
1310, 12imbi12d 344 . . . . . 6 (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
1413rspcv 3602 . . . . 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 2828 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wcel 2098  wral 3055  cdif 3940  cun 3941  wss 3943  𝒫 cpw 4597  {cpr 4625   cuni 4902   class class class wbr 5141  ran crn 5670  ωcom 7852  cdom 8939  sigAlgebracsiga 33636
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-2o 8468  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-dju 9898  df-card 9936  df-siga 33637
This theorem is referenced by:  measun  33739  aean  33772  sibfof  33869
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