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Theorem unelsiga 29524
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 4386 . . 3 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1072 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
3 isrnsigau 29517 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43simprd 478 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
54simp3d 1068 . . . 4 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
653ad2ant1 1075 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
7 prct 28875 . . . 4 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
873adant1 1072 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
9 prelpwi 4842 . . . . 5 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10 breq1 4586 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11 unieq 4380 . . . . . . . 8 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
1211eleq1d 2672 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → ( 𝑥𝑆 {𝐴, 𝐵} ∈ 𝑆))
1310, 12imbi12d 333 . . . . . 6 (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
1413rspcv 3278 . . . . 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 1072 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆) → ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
176, 8, 16mp2d 47 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝑆)
182, 17eqeltrrd 2689 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cdif 3537  cun 3538  wss 3540  𝒫 cpw 4108  {cpr 4127   cuni 4372   class class class wbr 4583  ran crn 5039  ωcom 6957  cdom 7839  sigAlgebracsiga 29497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-card 8648  df-cda 8873  df-siga 29498
This theorem is referenced by:  measun  29601  aean  29634  sibfof  29729
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