Theorem 0elsiga 31982

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 31981	. . 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 1146	. . . 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 1838	. 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 4047	. . . . . 6 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = (𝑜 ∖ 𝑜))
7		difid 4301	. . . . . 6 ⊢ (𝑜 ∖ 𝑜) = ∅
8	6, 7	eqtrdi 2795	. . . . 5 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = ∅)
9	8	eleq1d 2823	. . . 4 ⊢ (𝑥 = 𝑜 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
10	9	rspcva 3550	. . 3 ⊢ ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
11	10	exlimiv 1934	. 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 1085  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  ∪ cuni 4836   class class class wbr 5070  ran crn 5581  ωcom 7687   ≼ cdom 8689  sigAlgebracsiga 31976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426  df-siga 31977

This theorem is referenced by:  sigaclfu2  31989  sigaldsys  32027  brsiga  32051  measvuni  32082  measinb  32089  measres  32090  measdivcst  32092  measdivcstALTV  32093  cntmeas  32094  volmeas  32099  mbfmcst  32126  sibfof  32207  nuleldmp  32284  0rrv  32318  dstrvprob  32338