Theorem 0elsiga 31018

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 31017	. . 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 489	. 2 ⊢ (𝑆 ∈ ∪ ran sigAlgebra → ∃𝑜(𝑆 ⊆ 𝒫 𝑜 ∧ (𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝒫 𝑆(𝑥 ≼ ω → ∪ 𝑥 ∈ 𝑆))))
3		3simpa 1128	. . . 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 474	. . 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 1797	. 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 3985	. . . . . 6 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = (𝑜 ∖ 𝑜))
7		difid 4218	. . . . . 6 ⊢ (𝑜 ∖ 𝑜) = ∅
8	6, 7	syl6eq 2830	. . . . 5 ⊢ (𝑥 = 𝑜 → (𝑜 ∖ 𝑥) = ∅)
9	8	eleq1d 2850	. . . 4 ⊢ (𝑥 = 𝑜 → ((𝑜 ∖ 𝑥) ∈ 𝑆 ↔ ∅ ∈ 𝑆))
10	9	rspcva 3533	. . 3 ⊢ ((𝑜 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 (𝑜 ∖ 𝑥) ∈ 𝑆) → ∅ ∈ 𝑆)
11	10	exlimiv 1889	. 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 387   ∧ w3a 1068  ∃wex 1742   ∈ wcel 2050  ∀wral 3088  Vcvv 3415   ∖ cdif 3828   ⊆ wss 3831  ∅c0 4180  𝒫 cpw 4423  ∪ cuni 4713   class class class wbr 4930  ran crn 5409  ωcom 7398   ≼ cdom 8306  sigAlgebracsiga 31011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-fv 6198  df-siga 31012

This theorem is referenced by:  sigaclfu2  31025  sigaldsys  31063  brsiga  31087  measvuni  31118  measinb  31125  measres  31126  measdivcstOLD  31128  measdivcst  31129  cntmeas  31130  volmeas  31135  mbfmcst  31162  sibfof  31243  nuleldmp  31321  0rrv  31355  dstrvprob  31375