Theorem prsal 47092

Description: The pair of the empty set and the whole base is a sigma-algebra. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
prsal	⊢ (𝑋 ∈ 𝑉 → {∅, 𝑋} ∈ SAlg)

Proof of Theorem prsal

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 5272	. . . 4 ⊢ ∅ ∈ V
2	1	prid1 4730	. . 3 ⊢ ∅ ∈ {∅, 𝑋}
3	2	a1i 11	. 2 ⊢ (𝑋 ∈ 𝑉 → ∅ ∈ {∅, 𝑋})
4		uniprg 4890	. . . . . . . 8 ⊢ ((∅ ∈ V ∧ 𝑋 ∈ 𝑉) → ∪ {∅, 𝑋} = (∅ ∪ 𝑋))
5	1, 4	mpan 703	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ∪ {∅, 𝑋} = (∅ ∪ 𝑋))
6		0un 4353	. . . . . . 7 ⊢ (∅ ∪ 𝑋) = 𝑋
7	5, 6	eqtrdi 2816	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ∪ {∅, 𝑋} = 𝑋)
8	7	difeq1d 4080	. . . . 5 ⊢ (𝑋 ∈ 𝑉 → (∪ {∅, 𝑋} ∖ 𝑦) = (𝑋 ∖ 𝑦))
9	8	adantr 486	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) → (∪ {∅, 𝑋} ∖ 𝑦) = (𝑋 ∖ 𝑦))
10		difeq2 4075	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
11	10	adantl 487	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ∅) → (𝑋 ∖ 𝑦) = (𝑋 ∖ ∅))
12		dif0 4334	. . . . . . . 8 ⊢ (𝑋 ∖ ∅) = 𝑋
13	11, 12	eqtrdi 2816	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ∅) → (𝑋 ∖ 𝑦) = 𝑋)
14		prid2g 4729	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {∅, 𝑋})
15	14	adantr 486	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ∅) → 𝑋 ∈ {∅, 𝑋})
16	13, 15	eqeltrd 2865	. . . . . 6 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = ∅) → (𝑋 ∖ 𝑦) ∈ {∅, 𝑋})
17	16	adantlr 728	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) ∧ 𝑦 = ∅) → (𝑋 ∖ 𝑦) ∈ {∅, 𝑋})
18		neqne 2968	. . . . . . . 8 ⊢ (¬ 𝑦 = ∅ → 𝑦 ≠ ∅)
19		elprn1 4619	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, 𝑋} ∧ 𝑦 ≠ ∅) → 𝑦 = 𝑋)
20	18, 19	sylan2 605	. . . . . . 7 ⊢ ((𝑦 ∈ {∅, 𝑋} ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
21	20	adantll 727	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → 𝑦 = 𝑋)
22		difeq2 4075	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = (𝑋 ∖ 𝑋))
23		difid 4332	. . . . . . . 8 ⊢ (𝑋 ∖ 𝑋) = ∅
24	22, 23	eqtrdi 2816	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) = ∅)
25	2	a1i 11	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ∅ ∈ {∅, 𝑋})
26	24, 25	eqeltrd 2865	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑋 ∖ 𝑦) ∈ {∅, 𝑋})
27	21, 26	syl 18	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) ∧ ¬ 𝑦 = ∅) → (𝑋 ∖ 𝑦) ∈ {∅, 𝑋})
28	17, 27	pm2.61dan 825	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) → (𝑋 ∖ 𝑦) ∈ {∅, 𝑋})
29	9, 28	eqeltrd 2865	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {∅, 𝑋}) → (∪ {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
30	29	ralrimiva 3159	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ {∅, 𝑋} (∪ {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋})
31		elpwi 4571	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ⊆ {∅, 𝑋})
32	31	unissd 4884	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 {∅, 𝑋} → ∪ 𝑦 ⊆ ∪ {∅, 𝑋})
33	32	adantl 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) → ∪ 𝑦 ⊆ ∪ {∅, 𝑋})
34	7	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) → ∪ {∅, 𝑋} = 𝑋)
35	33, 34	sseqtrd 3974	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) → ∪ 𝑦 ⊆ 𝑋)
36	35	adantr 486	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋 ∈ 𝑦) → ∪ 𝑦 ⊆ 𝑋)
37		elssuni 4906	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑦 → 𝑋 ⊆ ∪ 𝑦)
38	37	adantl 487	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋 ∈ 𝑦) → 𝑋 ⊆ ∪ 𝑦)
39	36, 38	eqssd 3955	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋 ∈ 𝑦) → ∪ 𝑦 = 𝑋)
40	14	ad2antrr 739	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋 ∈ 𝑦) → 𝑋 ∈ {∅, 𝑋})
41	39, 40	eqeltrd 2865	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ 𝑋 ∈ 𝑦) → ∪ 𝑦 ∈ {∅, 𝑋})
42		id 23	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ 𝒫 {∅, 𝑋})
43		pwpr 4868	. . . . . . . . . . 11 ⊢ 𝒫 {∅, 𝑋} = ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}})
44	42, 43	eleqtrdi 2875	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 {∅, 𝑋} → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
45	44	adantr 486	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝒫 {∅, 𝑋} ∧ ¬ 𝑋 ∈ 𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
46	45	adantll 727	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → 𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}))
47		snidg 4628	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ {𝑋})
48	47	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {𝑋}) → 𝑋 ∈ {𝑋})
49		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {𝑋} → 𝑦 = {𝑋})
50	49	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {𝑋} → {𝑋} = 𝑦)
51	50	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {𝑋}) → {𝑋} = 𝑦)
52	48, 51	eleqtrd 2867	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {𝑋}) → 𝑋 ∈ 𝑦)
53	52	adantlr 728	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ 𝑦 = {𝑋}) → 𝑋 ∈ 𝑦)
54		id 23	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑉 → 𝑋 ∈ 𝑉)
55	54	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋 ∈ 𝑉)
56		neqne 2968	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 = {𝑋} → 𝑦 ≠ {𝑋})
57		elprn1 4619	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ 𝑦 ≠ {𝑋}) → 𝑦 = {∅, 𝑋})
58	56, 57	sylan2 605	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ {{𝑋}, {∅, 𝑋}} ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
59	58	adantll 727	. . . . . . . . . . . 12 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑦 = {∅, 𝑋})
60	14	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {∅, 𝑋}) → 𝑋 ∈ {∅, 𝑋})
61		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = {∅, 𝑋} → 𝑦 = {∅, 𝑋})
62	61	eqcomd 2771	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {∅, 𝑋} → {∅, 𝑋} = 𝑦)
63	62	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {∅, 𝑋}) → {∅, 𝑋} = 𝑦)
64	60, 63	eleqtrd 2867	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 = {∅, 𝑋}) → 𝑋 ∈ 𝑦)
65	55, 59, 64	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 = {𝑋}) → 𝑋 ∈ 𝑦)
66	53, 65	pm2.61dan 825	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑋 ∈ 𝑦)
67	66	stoic1a 1805	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
68	67	adantlr 728	. . . . . . . 8 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}})
69		elunnel2 4109	. . . . . . . 8 ⊢ ((𝑦 ∈ ({∅, {∅}} ∪ {{𝑋}, {∅, 𝑋}}) ∧ ¬ 𝑦 ∈ {{𝑋}, {∅, 𝑋}}) → 𝑦 ∈ {∅, {∅}})
70	46, 68, 69	syl2anc 596	. . . . . . 7 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → 𝑦 ∈ {∅, {∅}})
71		unieq 4885	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∪ ∅)
72		uni0 4903	. . . . . . . . . 10 ⊢ ∪ ∅ = ∅
73	71, 72	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ∪ 𝑦 = ∅)
74	73	adantl 487	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 = ∅) → ∪ 𝑦 = ∅)
75		elprn1 4619	. . . . . . . . . 10 ⊢ ((𝑦 ∈ {∅, {∅}} ∧ 𝑦 ≠ ∅) → 𝑦 = {∅})
76	18, 75	sylan2 605	. . . . . . . . 9 ⊢ ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → 𝑦 = {∅})
77		unieq 4885	. . . . . . . . . 10 ⊢ (𝑦 = {∅} → ∪ 𝑦 = ∪ {∅})
78		unisn0 45834	. . . . . . . . . 10 ⊢ ∪ {∅} = ∅
79	77, 78	eqtrdi 2816	. . . . . . . . 9 ⊢ (𝑦 = {∅} → ∪ 𝑦 = ∅)
80	76, 79	syl 18	. . . . . . . 8 ⊢ ((𝑦 ∈ {∅, {∅}} ∧ ¬ 𝑦 = ∅) → ∪ 𝑦 = ∅)
81	74, 80	pm2.61dan 825	. . . . . . 7 ⊢ (𝑦 ∈ {∅, {∅}} → ∪ 𝑦 = ∅)
82	70, 81	syl 18	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → ∪ 𝑦 = ∅)
83	2	a1i 11	. . . . . 6 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → ∅ ∈ {∅, 𝑋})
84	82, 83	eqeltrd 2865	. . . . 5 ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) ∧ ¬ 𝑋 ∈ 𝑦) → ∪ 𝑦 ∈ {∅, 𝑋})
85	41, 84	pm2.61dan 825	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) → ∪ 𝑦 ∈ {∅, 𝑋})
86	85	a1d 26	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑦 ∈ 𝒫 {∅, 𝑋}) → (𝑦 ≼ ω → ∪ 𝑦 ∈ {∅, 𝑋}))
87	86	ralrimiva 3159	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → ∪ 𝑦 ∈ {∅, 𝑋}))
88		prex 5411	. . 3 ⊢ {∅, 𝑋} ∈ V
89		issal 47088	. . 3 ⊢ ({∅, 𝑋} ∈ V → ({∅, 𝑋} ∈ SAlg ↔ (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} (∪ {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → ∪ 𝑦 ∈ {∅, 𝑋}))))
90	88, 89	mp1i 14	. 2 ⊢ (𝑋 ∈ 𝑉 → ({∅, 𝑋} ∈ SAlg ↔ (∅ ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ {∅, 𝑋} (∪ {∅, 𝑋} ∖ 𝑦) ∈ {∅, 𝑋} ∧ ∀𝑦 ∈ 𝒫 {∅, 𝑋} (𝑦 ≼ ω → ∪ 𝑦 ∈ {∅, 𝑋}))))
91	3, 30, 87, 90	mpbir3and 1361	1 ⊢ (𝑋 ∈ 𝑉 → {∅, 𝑋} ∈ SAlg)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  𝒫 cpw 4564  {csn 4591  {cpr 4593  ∪ cuni 4874   class class class wbr 5111  ωcom 7868   ≼ cdom 8947  SAlgcsalg 47082

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-pw 4566  df-sn 4592  df-pr 4594  df-uni 4875  df-salg 47083

This theorem is used by: (None)