Theorem 0elsiga 34087

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.

Step	Hyp	Ref	Expression
1		isrnsiga 34086	. . 3 ⊢ (𝑆 ∈ ∪ ran sigAlgebra ↔ (𝑆 ∈ V ∧ ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)))))
2	1	simprbi 496	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
3		3simpa 1148	. . . 4 ⊢ ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆)) → (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆))
4	3	adantl 481	. . 3 ⊢ ((𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆))
5	4	eximi 1835	. 2 ⊢ (∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))) → ∃𝑜(𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆))
6		difeq2 4071	. . . . . 6 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = (𝑜 ∖ 𝑜))
7		difid 4327	. . . . . 6 ⊢ (𝑜 ∖ 𝑜) = ∅
8	6, 7	eqtrdi 2780	. . . . 5 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = ∅)
9	8	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑜 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
10	9	rspcva 3575	. . 3 ⊢ ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
11	10	exlimiv 1930	. 2 ⊢ (∃𝑜(𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
12	2, 5, 11	3syl 18	1 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∅ ∈ 𝑆)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4284  𝒫 cpw 4551  ∪ cuni 4858   class class class wbr 5092  ran crn 5620  ωcom 7799   ≼ cdom 8870  sigAlgebracsiga 34081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-fv 6490  df-siga 34082

This theorem is referenced by:  sigaclfu2  34094  sigaldsys  34132  brsiga  34156  measvuni  34187  measinb  34194  measres  34195  measdivcst  34197  measdivcstALTV  34198  cntmeas  34199  volmeas  34204  mbfmcst  34233  sibfof  34314  nuleldmp  34391  0rrv  34425  dstrvprob  34446