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Theorem 0elsiga 30305
Description: A sigma-algebra contains the empty set. (Contributed by Thierry Arnoux, 4-Sep-2016.)
Assertion
Ref Expression
0elsiga (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)

Proof of Theorem 0elsiga
Dummy variables 𝑜 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isrnsiga 30304 . . 3 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
21simprbi 479 . 2 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3 3simpa 1078 . . . 4 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
43adantl 481 . . 3 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
54eximi 1802 . 2 (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∃𝑜(𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
6 difeq2 3755 . . . . . 6 (𝑥 = 𝑜 → (𝑜𝑥) = (𝑜𝑜))
7 difid 3981 . . . . . 6 (𝑜𝑜) = ∅
86, 7syl6eq 2701 . . . . 5 (𝑥 = 𝑜 → (𝑜𝑥) = ∅)
98eleq1d 2715 . . . 4 (𝑥 = 𝑜 → ((𝑜𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
109rspcva 3338 . . 3 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
1110exlimiv 1898 . 2 (∃𝑜(𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
122, 5, 113syl 18 1 (𝑆 ran sigAlgebra → ∅ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054  wex 1744  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  wss 3607  c0 3948  𝒫 cpw 4191   cuni 4468   class class class wbr 4685  ran crn 5144  ωcom 7107  cdom 7995  sigAlgebracsiga 30298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-siga 30299
This theorem is referenced by:  sigaclfu2  30312  sigaldsys  30350  brsiga  30374  measvuni  30405  measinb  30412  measres  30413  measdivcstOLD  30415  measdivcst  30416  cntmeas  30417  volmeas  30422  mbfmcst  30449  sibfof  30530  nuleldmp  30607  0rrv  30641  dstrvprob  30661
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