Theorem 0elsiga 34115

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 34114	. . 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 1149	. . . 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 4120	. . . . . 6 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = (𝑜 ∖ 𝑜))
7		difid 4376	. . . . . 6 ⊢ (𝑜 ∖ 𝑜) = ∅
8	6, 7	eqtrdi 2793	. . . . 5 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = ∅)
9	8	eleq1d 2826	. . . 4 ⊢ (𝑥 = 𝑜 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
10	9	rspcva 3620	. . 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 1087  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  ∪ cuni 4907   class class class wbr 5143  ran crn 5686  ωcom 7887   ≼ cdom 8983  sigAlgebracsiga 34109

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-siga 34110

This theorem is referenced by:  sigaclfu2  34122  sigaldsys  34160  brsiga  34184  measvuni  34215  measinb  34222  measres  34223  measdivcst  34225  measdivcstALTV  34226  cntmeas  34227  volmeas  34232  mbfmcst  34261  sibfof  34342  nuleldmp  34419  0rrv  34453  dstrvprob  34474