Theorem prsiga 29521

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 4760	. . 3 ⊢ ∅ ∈ 𝒫 𝑂
2		pwidg 4121	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ 𝒫 𝑂)
3		prssi 4293	. . 3 ⊢ ((∅ ∈ 𝒫 𝑂 ∧ 𝑂 ∈ 𝒫 𝑂) → {∅, 𝑂} ⊆ 𝒫 𝑂)
4	1, 2, 3	sylancr 694	. 2 ⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ⊆ 𝒫 𝑂)
5		prid2g 4240	. . 3 ⊢ (𝑂 ∈ 𝑉 → 𝑂 ∈ {∅, 𝑂})
6		dif0 3904	. . . . 5 ⊢ (𝑂 ∖ ∅) = 𝑂
7	6, 5	syl5eqel 2692	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ ∅) ∈ {∅, 𝑂})
8		difid 3902	. . . . 5 ⊢ (𝑂 ∖ 𝑂) = ∅
9		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
10	9	prid1 4241	. . . . . 6 ⊢ ∅ ∈ {∅, 𝑂}
11	10	a1i 11	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → ∅ ∈ {∅, 𝑂})
12	8, 11	syl5eqel 2692	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})
13		difeq2 3684	. . . . . . 7 ⊢ (𝑥 = ∅ → (𝑂 ∖ 𝑥) = (𝑂 ∖ ∅))
14	13	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = ∅ → ((𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ ∅) ∈ {∅, 𝑂}))
15		difeq2 3684	. . . . . . 7 ⊢ (𝑥 = 𝑂 → (𝑂 ∖ 𝑥) = (𝑂 ∖ 𝑂))
16	15	eleq1d 2672	. . . . . 6 ⊢ (𝑥 = 𝑂 → ((𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂}))
17	14, 16	ralprg 4181	. . . . 5 ⊢ ((∅ ∈ V ∧ 𝑂 ∈ 𝑉) → (∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})))
18	9, 17	mpan 702	. . . 4 ⊢ (𝑂 ∈ 𝑉 → (∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ↔ ((𝑂 ∖ ∅) ∈ {∅, 𝑂} ∧ (𝑂 ∖ 𝑂) ∈ {∅, 𝑂})))
19	7, 12, 18	mpbir2and 959	. . 3 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂})
20		uni0 4401	. . . . . . . . 9 ⊢ ∪ ∅ = ∅
21	20, 10	eqeltri 2684	. . . . . . . 8 ⊢ ∪ ∅ ∈ {∅, 𝑂}
22	9	unisn 4387	. . . . . . . . 9 ⊢ ∪ {∅} = ∅
23	22, 10	eqeltri 2684	. . . . . . . 8 ⊢ ∪ {∅} ∈ {∅, 𝑂}
24	21, 23	pm3.2i 470	. . . . . . 7 ⊢ (∪ ∅ ∈ {∅, 𝑂} ∧ ∪ {∅} ∈ {∅, 𝑂})
25		snex 4835	. . . . . . . . 9 ⊢ {∅} ∈ V
26	9, 25	pm3.2i 470	. . . . . . . 8 ⊢ (∅ ∈ V ∧ {∅} ∈ V)
27		unieq 4380	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → ∪ 𝑥 = ∪ ∅)
28	27	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ ∅ ∈ {∅, 𝑂}))
29		unieq 4380	. . . . . . . . . 10 ⊢ (𝑥 = {∅} → ∪ 𝑥 = ∪ {∅})
30	29	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = {∅} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {∅} ∈ {∅, 𝑂}))
31	28, 30	ralprg 4181	. . . . . . . 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 247	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂})
34		unisng 4388	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → ∪ {𝑂} = 𝑂)
35	34, 5	eqeltrd 2688	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → ∪ {𝑂} ∈ {∅, 𝑂})
36		uniprg 4386	. . . . . . . . . 10 ⊢ ((∅ ∈ V ∧ 𝑂 ∈ 𝑉) → ∪ {∅, 𝑂} = (∅ ∪ 𝑂))
37	9, 36	mpan 702	. . . . . . . . 9 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} = (∅ ∪ 𝑂))
38		uncom 3719	. . . . . . . . . 10 ⊢ (∅ ∪ 𝑂) = (𝑂 ∪ ∅)
39		un0 3919	. . . . . . . . . 10 ⊢ (𝑂 ∪ ∅) = 𝑂
40	38, 39	eqtri 2632	. . . . . . . . 9 ⊢ (∅ ∪ 𝑂) = 𝑂
41	37, 40	syl6eq 2660	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} = 𝑂)
42	41, 5	eqeltrd 2688	. . . . . . 7 ⊢ (𝑂 ∈ 𝑉 → ∪ {∅, 𝑂} ∈ {∅, 𝑂})
43		snex 4835	. . . . . . . . 9 ⊢ {𝑂} ∈ V
44		prex 4836	. . . . . . . . 9 ⊢ {∅, 𝑂} ∈ V
45	43, 44	pm3.2i 470	. . . . . . . 8 ⊢ ({𝑂} ∈ V ∧ {∅, 𝑂} ∈ V)
46		unieq 4380	. . . . . . . . . 10 ⊢ (𝑥 = {𝑂} → ∪ 𝑥 = ∪ {𝑂})
47	46	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = {𝑂} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {𝑂} ∈ {∅, 𝑂}))
48		unieq 4380	. . . . . . . . . 10 ⊢ (𝑥 = {∅, 𝑂} → ∪ 𝑥 = ∪ {∅, 𝑂})
49	48	eleq1d 2672	. . . . . . . . 9 ⊢ (𝑥 = {∅, 𝑂} → (∪ 𝑥 ∈ {∅, 𝑂} ↔ ∪ {∅, 𝑂} ∈ {∅, 𝑂}))
50	47, 49	ralprg 4181	. . . . . . . 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 959	. . . . . 6 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂})
53		ralun 3757	. . . . . 6 ⊢ ((∀𝑥 ∈ {∅, {∅}}∪ 𝑥 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {{𝑂}, {∅, 𝑂}}∪ 𝑥 ∈ {∅, 𝑂}) → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
54	33, 52, 53	syl2anc 691	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
55		pwpr 4368	. . . . . 6 ⊢ 𝒫 {∅, 𝑂} = ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})
56	55	raleqi 3119	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 {∅, 𝑂}∪ 𝑥 ∈ {∅, 𝑂} ↔ ∀𝑥 ∈ ({∅, {∅}} ∪ {{𝑂}, {∅, 𝑂}})∪ 𝑥 ∈ {∅, 𝑂})
57	54, 56	sylibr 223	. . . 4 ⊢ (𝑂 ∈ 𝑉 → ∀𝑥 ∈ 𝒫 {∅, 𝑂}∪ 𝑥 ∈ {∅, 𝑂})
58		ax-1 6	. . . . 5 ⊢ (∪ 𝑥 ∈ {∅, 𝑂} → (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂}))
59	58	ralimi 2936	. . . 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 1235	. 2 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ {∅, 𝑂} (𝑂 ∖ 𝑥) ∈ {∅, 𝑂} ∧ ∀𝑥 ∈ 𝒫 {∅, 𝑂} (𝑥 ≼ ω → ∪ 𝑥 ∈ {∅, 𝑂})))
62		issiga 29501	. . 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 695	1 ⊢ (𝑂 ∈ 𝑉 → {∅, 𝑂} ∈ (sigAlgebra‘𝑂))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804  ωcom 6957   ≼ cdom 7839  sigAlgebracsiga 29497

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-siga 29498

This theorem is referenced by: (None)