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Theorem 0elsiga 31495
 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 31494 . . 3 (𝑆 ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)))))
21simprbi 500 . 2 (𝑆 ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))))
3 3simpa 1145 . . . 4 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆)) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
43adantl 485 . . 3 ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
54eximi 1836 . 2 (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → 𝑥𝑆))) → ∃𝑜(𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆))
6 difeq2 4044 . . . . . 6 (𝑥 = 𝑜 → (𝑜𝑥) = (𝑜𝑜))
7 difid 4284 . . . . . 6 (𝑜𝑜) = ∅
86, 7eqtrdi 2849 . . . . 5 (𝑥 = 𝑜 → (𝑜𝑥) = ∅)
98eleq1d 2874 . . . 4 (𝑥 = 𝑜 → ((𝑜𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
109rspcva 3569 . . 3 ((𝑜𝑆 ∧ ∀𝑥𝑆 (𝑜𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
1110exlimiv 1931 . 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 399   ∧ w3a 1084  ∃wex 1781   ∈ wcel 2111  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4800   class class class wbr 5030  ran crn 5520  ωcom 7562   ≼ cdom 8492  sigAlgebracsiga 31489 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332  df-siga 31490 This theorem is referenced by:  sigaclfu2  31502  sigaldsys  31540  brsiga  31564  measvuni  31595  measinb  31602  measres  31603  measdivcst  31605  measdivcstALTV  31606  cntmeas  31607  volmeas  31612  mbfmcst  31639  sibfof  31720  nuleldmp  31797  0rrv  31831  dstrvprob  31851
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