Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  unelsiga Structured version   Visualization version   GIF version

Theorem unelsiga 33786
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 4928 . . 3 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
213adant1 1127 . 2 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} = (𝐴𝐵))
3 isrnsigau 33779 . . . . . 6 (𝑆 ran sigAlgebra → (𝑆 ⊆ 𝒫 𝑆 ∧ ( 𝑆𝑆 ∧ ∀𝑥𝑆 ( 𝑆𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
43simprd 494 . . . . 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 32517 . . . 4 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
873adant1 1127 . . 3 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ≼ ω)
9 prelpwi 5453 . . . . 5 ((𝐴𝑆𝐵𝑆) → {𝐴, 𝐵} ∈ 𝒫 𝑆)
10 breq1 5155 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → (𝑥 ≼ ω ↔ {𝐴, 𝐵} ≼ ω))
11 unieq 4923 . . . . . . . 8 (𝑥 = {𝐴, 𝐵} → 𝑥 = {𝐴, 𝐵})
1211eleq1d 2814 . . . . . . 7 (𝑥 = {𝐴, 𝐵} → ( 𝑥𝑆 {𝐴, 𝐵} ∈ 𝑆))
1310, 12imbi12d 343 . . . . . 6 (𝑥 = {𝐴, 𝐵} → ((𝑥 ≼ ω → 𝑥𝑆) ↔ ({𝐴, 𝐵} ≼ ω → {𝐴, 𝐵} ∈ 𝑆)))
1413rspcv 3607 . . . . 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 2830 1 ((𝑆 ran sigAlgebra ∧ 𝐴𝑆𝐵𝑆) → (𝐴𝐵) ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3058  cdif 3946  cun 3947  wss 3949  𝒫 cpw 4606  {cpr 4634   cuni 4912   class class class wbr 5152  ran crn 5683  ωcom 7876  cdom 8968  sigAlgebracsiga 33760
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-inf2 9672
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-om 7877  df-1st 7999  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-2o 8494  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-oi 9541  df-dju 9932  df-card 9970  df-siga 33761
This theorem is referenced by:  measun  33863  aean  33896  sibfof  33993
  Copyright terms: Public domain W3C validator