Theorem prsiga 31999

Description: The smallest possible sigma-algebra containing 𝑂. (Contributed by Thierry Arnoux, 13-Sep-2016.)

Assertion

Ref	Expression
prsiga	⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))

Proof of Theorem prsiga

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0elpw 5273	. . 3 ⊢ ∅ ∈ 𝒫 𝑂
2		pwidg 4552	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
3		prssi 4751	. . 3 ⊢ ((∅ ∈ 𝒫 𝑂 ∧ 𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
4	1, 2, 3	sylancr 586	. 2 ⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5		prid2g 4694	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ {∅, 𝑂})
6		dif0 4303	. . . . 5 ⊢ (𝑂 ∖ ∅) = 𝑂
7	6, 5	eqeltrid 2843	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8		difid 4301	. . . . 5 ⊢ (𝑂 ∖ 𝑂) = ∅
9		0ex 5226	. . . . . . 7 ⊢ ∅ ∈ V
10	9	prid1 4695	. . . . . 6 ⊢ ∅ ∈ {∅, 𝑂}
11	10	a1i 11	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ {∅, 𝑂})
12	8, 11	eqeltrid 2843	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})
13		difeq2 4047	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑂 ∖ 𝑥) = (𝑂 ∖ ∅))
14	13	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15		difeq2 4047	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑂 ∖ 𝑥) = (𝑂 ∖ 𝑂))
16	15	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂}))
17	14, 16	ralprg 4627	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑂 ∈ 𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})))
18	9, 17	mpan 686	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})))
19	7, 12, 18	mpbir2and 709	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂})
20		uni0 4866	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
21	20, 10	eqeltri 2835	. . . . . . . 8 ⊢ ∪ ∅ ∈ {∅, 𝑂}
22	9	unisn 4858	. . . . . . . . 9 ⊢ ∪ {∅} = ∅
23	22, 10	eqeltri 2835	. . . . . . . 8 ⊢ ∪ {∅} ∈ {∅, 𝑂}
24	21, 23	pm3.2i 470	. . . . . . 7 ⊢ (∪ ∅ ∈ {∅, 𝑂} ∧ ∪ {∅} ∈ {∅, 𝑂})
25		snex 5349	. . . . . . . . 9 ⊢ {∅} ∈ V
26	9, 25	pm3.2i 470	. . . . . . . 8 ⊢ (∅ ∈ V ∧ {∅} ∈ V)
27		unieq 4847	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
28	27	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ ∅ ∈ {∅, 𝑂}))
29		unieq 4847	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → ∪ 𝑥 = ∪ {∅})
30	29	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = {∅} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {∅} ∈ {∅, 𝑂}))
31	28, 30	ralprg 4627	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ {∅} ∈ V) → (∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂} ↔ (∪ ∅ ∈ {∅, 𝑂} ∧ ∪ {∅} ∈ {∅, 𝑂})))
32	26, 31	mp1i 13	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂} ↔ (∪ ∅ ∈ {∅, 𝑂} ∧ ∪ {∅} ∈ {∅, 𝑂})))
33	24, 32	mpbiri 257	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂})
34		unisng 4857	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → ∪ {𝑂} = 𝑂)
35	34, 5	eqeltrd 2839	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → ∪ {𝑂} ∈ {∅, 𝑂})
36		uniprg 4853	. . . . . . . . . 10 ⊢ ((∅ ∈ V ∧ 𝑂 ∈ 𝑉) → ∪ {∅, 𝑂} = (∅ ∪ 𝑂))
37	9, 36	mpan 686	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} = (∅ ∪ 𝑂))
38		uncom 4083	. . . . . . . . . 10 ⊢ (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39		un0 4321	. . . . . . . . . 10 ⊢ (𝑂 ∪ ∅) = 𝑂
40	38, 39	eqtri 2766	. . . . . . . . 9 ⊢ (∅ ∪ 𝑂) = 𝑂
41	37, 40	eqtrdi 2795	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} = 𝑂)
42	41, 5	eqeltrd 2839	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} ∈ {∅, 𝑂})
43		snex 5349	. . . . . . . . 9 ⊢ {𝑂} ∈ V
44		prex 5350	. . . . . . . . 9 ⊢ {∅, 𝑂} ∈ V
45	43, 44	pm3.2i 470	. . . . . . . 8 ⊢ ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46		unieq 4847	. . . . . . . . . 10 ⊢ (𝑥 = {𝑂} → ∪ 𝑥 = ∪ {𝑂})
47	46	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = {𝑂} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {𝑂} ∈ {∅, 𝑂}))
48		unieq 4847	. . . . . . . . . 10 ⊢ (𝑥 = {∅, 𝑂} → ∪ 𝑥 = ∪ {∅, 𝑂})
49	48	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = {∅, 𝑂} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {∅, 𝑂} ∈ {∅, 𝑂}))
50	47, 49	ralprg 4627	. . . . . . . 8 ⊢ (({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V) → (∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂} ↔ (∪ {𝑂} ∈ {∅, 𝑂} ∧ ∪ {∅, 𝑂} ∈ {∅, 𝑂})))
51	45, 50	mp1i 13	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → (∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂} ↔ (∪ {𝑂} ∈ {∅, 𝑂} ∧ ∪ {∅, 𝑂} ∈ {∅, 𝑂})))
52	35, 42, 51	mpbir2and 709	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂})
53		ralun 4122	. . . . . 6 ⊢ ((∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂}) → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
54	33, 52, 53	syl2anc 583	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
55		pwpr 4830	. . . . . 6 ⊢ 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
56	55	raleqi 3337	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 {∅, 𝑂}∪ 𝑥 ∈ {∅, 𝑂} ↔ ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
57	54, 56	sylibr 233	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂}∪ 𝑥 ∈ {∅, 𝑂})
58		ax-1 6	. . . . 5 ⊢ (∪ 𝑥 ∈ {∅, 𝑂} → (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂}))
59	58	ralimi 3086	. . . 4 ⊢ (∀𝑥 ∈ 𝒫 {∅, 𝑂}∪ 𝑥 ∈ {∅, 𝑂} → ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂}))
60	57, 59	syl 17	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂}))
61	5, 19, 60	3jca 1126	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂})))
62		issiga 31980	. . 3 ⊢ ({∅, 𝑂} ∈ V → ({∅, 𝑂} ∈ (sigAlgebra‘𝑂) ↔ ({∅, 𝑂} ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂})))))
63	44, 62	ax-mp 5	. 2 ⊢ ({∅, 𝑂} ∈ (sigAlgebra‘𝑂) ↔ ({∅, 𝑂} ⊆ 𝒫 𝑂 ∧ (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂}))))
64	4, 61, 63	sylanbrc 582	1 ⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  Vcvv 3422   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  𝒫 cpw 4530  {csn 4558  {cpr 4560  ∪ cuni 4836   class class class wbr 5070  ‘cfv 6418  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-iota 6376  df-fun 6420  df-fv 6426  df-siga 31977

This theorem is referenced by: (None)