Theorem 0elsiga 34078

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 34077	. . 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 1833	. 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 4143	. . . . . 6 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = (𝑜 ∖ 𝑜))
7		difid 4398	. . . . . 6 ⊢ (𝑜 ∖ 𝑜) = ∅
8	6, 7	eqtrdi 2796	. . . . 5 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = ∅)
9	8	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝑜 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
10	9	rspcva 3633	. . 3 ⊢ ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
11	10	exlimiv 1929	. 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 1087  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∖ cdif 3973   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  ran crn 5701  ωcom 7903   ≼ cdom 9001  sigAlgebracsiga 34072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-siga 34073

This theorem is referenced by:  sigaclfu2  34085  sigaldsys  34123  brsiga  34147  measvuni  34178  measinb  34185  measres  34186  measdivcst  34188  measdivcstALTV  34189  cntmeas  34190  volmeas  34195  mbfmcst  34224  sibfof  34305  nuleldmp  34382  0rrv  34416  dstrvprob  34436