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Theorem unelsiga 30504
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 4600 . . 3 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1125 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
3 isrnsigau 30497 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43simprd 482 . . . . 5 (𝑆 ran sigAlgebra → ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))
54simp3d 1139 . . . 4 (𝑆 ran sigAlgebra → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
653ad2ant1 1128 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))
7 prct 29799 . . . 4 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
873adant1 1125 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
9 prelpwi 5062 . . . . 5 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10 breq1 4805 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11 unieq 4594 . . . . . . . 8 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
1211eleq1d 2822 . . . . . . 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 3443 . . . . 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 1125 . . 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 2838 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1630  wcel 2137  wral 3048  cdif 3710  cun 3711  wss 3713  𝒫 cpw 4300  {cpr 4321   cuni 4586   class class class wbr 4802  ran crn 5265  ωcom 7228  cdom 8117  sigAlgebracsiga 30477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112  ax-inf2 8709
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rmo 3056  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-se 5224  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-pred 5839  df-ord 5885  df-on 5886  df-lim 5887  df-suc 5888  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-isom 6056  df-riota 6772  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-om 7229  df-1st 7331  df-2nd 7332  df-wrecs 7574  df-recs 7635  df-rdg 7673  df-1o 7727  df-2o 7728  df-oadd 7731  df-er 7909  df-en 8120  df-dom 8121  df-sdom 8122  df-fin 8123  df-oi 8578  df-card 8953  df-cda 9180  df-siga 30478
This theorem is referenced by:  measun  30581  aean  30614  sibfof  30709
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