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Theorem 0elsiga 33181
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 33180 . . 3 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
21simprbi 497 . 2 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3 3simpa 1148 . . . 4 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
43adantl 482 . . 3 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
54eximi 1837 . 2 (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∃𝑜(𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
6 difeq2 4116 . . . . . 6 (𝑥 = 𝑜 → (𝑜𝑥) = (𝑜𝑜))
7 difid 4370 . . . . . 6 (𝑜𝑜) = ∅
86, 7eqtrdi 2788 . . . . 5 (𝑥 = 𝑜 → (𝑜𝑥) = ∅)
98eleq1d 2818 . . . 4 (𝑥 = 𝑜 → ((𝑜𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
109rspcva 3610 . . 3 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
1110exlimiv 1933 . 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 396  w3a 1087  wex 1781  wcel 2106  wral 3061  Vcvv 3474  cdif 3945  wss 3948  c0 4322  𝒫 cpw 4602   cuni 4908   class class class wbr 5148  ran crn 5677  ωcom 7857  cdom 8939  sigAlgebracsiga 33175
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-siga 33176
This theorem is referenced by:  sigaclfu2  33188  sigaldsys  33226  brsiga  33250  measvuni  33281  measinb  33288  measres  33289  measdivcst  33291  measdivcstALTV  33292  cntmeas  33293  volmeas  33298  mbfmcst  33327  sibfof  33408  nuleldmp  33485  0rrv  33519  dstrvprob  33539
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