ldsysgenld - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > ldsysgenld		Structured version Visualization version GIF version

Theorem ldsysgenld 33453

Description: The intersection of all lambda-systems containing a given collection of sets 𝐴, which is called the lambda-system generated by 𝐴, is itself also a lambda-system. (Contributed by Thierry Arnoux, 16-Jun-2020.)

**Hypotheses**
Ref	Expression
isldsys.l	⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
ldsysgenld.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
ldsysgenld.2	⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)

**Assertion**
Ref	Expression
ldsysgenld	⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝐿)

Distinct variable groups: 𝑦,𝑠 𝑡,𝐿 𝑂,𝑠,𝑡,𝑥 𝑥,𝑉 𝑦,𝑡 𝐴,𝑠,𝑡,𝑥 𝐿,𝑠,𝑥 𝜑,𝑡,𝑥 Allowed substitution hints: 𝜑(𝑦,𝑠) 𝐴(𝑦) 𝐿(𝑦) 𝑂(𝑦) 𝑉(𝑦,𝑡,𝑠)

Proof of Theorem ldsysgenld Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ldsysgenld.1	. . . . 5 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
2		pwsiga 33423	. . . . 5 ⊢ (𝑂 ∈ 𝑉 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → 𝒫 𝑂 ∈ (sigAlgebra‘𝑂))
4		isldsys.l	. . . . . . . 8 ⊢ 𝐿 = {𝑠 ∈ 𝒫 𝒫 𝑂 ∣ (∅ ∈ 𝑠 ∧ ∀𝑥 ∈ 𝑠 (𝑂 ∖ 𝑥) ∈ 𝑠 ∧ ∀𝑥 ∈ 𝒫 𝑠((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑠))}
5	4	sigaldsys 33452	. . . . . . 7 ⊢ (sigAlgebra‘𝑂) ⊆ 𝐿
6	5, 3	sselid 3981	. . . . . 6 ⊢ (𝜑 → 𝒫 𝑂 ∈ 𝐿)
7		ldsysgenld.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝒫 𝑂)
8		sseq2 4009	. . . . . . 7 ⊢ (𝑡 = 𝒫 𝑂 → (𝐴 ⊆ 𝑡 ↔ 𝐴 ⊆ 𝒫 𝑂))
9	8	elrab 3684	. . . . . 6 ⊢ (𝒫 𝑂 ∈ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ (𝒫 𝑂 ∈ 𝐿 ∧ 𝐴 ⊆ 𝒫 𝑂))
10	6, 7, 9	sylanbrc 582	. . . . 5 ⊢ (𝜑 → 𝒫 𝑂 ∈ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
11		intss1 4968	. . . . 5 ⊢ (𝒫 𝑂 ∈ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ⊆ 𝒫 𝑂)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ⊆ 𝒫 𝑂)
13	3, 12	sselpwd 5327	. . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝒫 𝒫 𝑂)
14	4	isldsys 33449	. . . . . . . . . 10 ⊢ (𝑡 ∈ 𝐿 ↔ (𝑡 ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))))
15	14	simprbi 496	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐿 → (∅ ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡)))
16	15	simp1d 1141	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐿 → ∅ ∈ 𝑡)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → ∅ ∈ 𝑡)
18	17	a1d 25	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐿) → (𝐴 ⊆ 𝑡 → ∅ ∈ 𝑡))
19	18	ralrimiva 3145	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → ∅ ∈ 𝑡))
20		0ex 5308	. . . . . 6 ⊢ ∅ ∈ V
21	20	elintrab 4965	. . . . 5 ⊢ (∅ ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → ∅ ∈ 𝑡))
22	19, 21	sylibr 233	. . . 4 ⊢ (𝜑 → ∅ ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
23		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑡𝜑
24		nfcv 2902	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
25		nfrab1 3450	. . . . . . . . . 10 ⊢ Ⅎ𝑡{𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}
26	25	nfint 4961	. . . . . . . . 9 ⊢ Ⅎ𝑡∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}
27	24, 26	nfel 2916	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}
28	23, 27	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
29		simplr 766	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑡 ∈ 𝐿)
30		vex 3477	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
31	30	elintrab 4965	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → 𝑥 ∈ 𝑡))
32	31	biimpi 215	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → 𝑥 ∈ 𝑡))
33	32	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → 𝑥 ∈ 𝑡))
34	33	r19.21bi 3247	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ 𝑡 ∈ 𝐿) → (𝐴 ⊆ 𝑡 → 𝑥 ∈ 𝑡))
35	34	imp 406	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑥 ∈ 𝑡)
36	15	simp2d 1142	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝐿 → ∀𝑥 ∈ 𝑡 (𝑂 ∖ 𝑥) ∈ 𝑡)
37	36	r19.21bi 3247	. . . . . . . . . 10 ⊢ ((𝑡 ∈ 𝐿 ∧ 𝑥 ∈ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
38	29, 35, 37	syl2anc 583	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → (𝑂 ∖ 𝑥) ∈ 𝑡)
39	38	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ 𝑡 ∈ 𝐿) → (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡))
40	39	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → (𝑡 ∈ 𝐿 → (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡)))
41	28, 40	ralrimi 3253	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡))
42		difexg 5328	. . . . . . . 8 ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∖ 𝑥) ∈ V)
43		elintrabg 4966	. . . . . . . 8 ⊢ ((𝑂 ∖ 𝑥) ∈ V → ((𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡)))
44	1, 42, 43	3syl 18	. . . . . . 7 ⊢ (𝜑 → ((𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡)))
45	44	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → ((𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → (𝑂 ∖ 𝑥) ∈ 𝑡)))
46	41, 45	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → (𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
47	46	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} (𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
48	26	nfpw 4622	. . . . . . . . . . 11 ⊢ Ⅎ𝑡𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}
49	24, 48	nfel 2916	. . . . . . . . . 10 ⊢ Ⅎ𝑡 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}
50	23, 49	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
51		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)
52	50, 51	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦))
53		simplr 766	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑡 ∈ 𝐿)
54		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) ∧ 𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → 𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
55		simpllr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) ∧ 𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → 𝑡 ∈ 𝐿)
56		simplr 766	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) ∧ 𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → 𝐴 ⊆ 𝑡)
57		vex 3477	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑢 ∈ V
58	57	elintrab 4965	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → 𝑢 ∈ 𝑡))
59	58	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → 𝑢 ∈ 𝑡))
60	59	r19.21bi 3247	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ 𝑡 ∈ 𝐿) → (𝐴 ⊆ 𝑡 → 𝑢 ∈ 𝑡))
61	60	imp 406	. . . . . . . . . . . . . . . 16 ⊢ (((𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑢 ∈ 𝑡)
62	54, 55, 56, 61	syl21anc 835	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) ∧ 𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → 𝑢 ∈ 𝑡)
63	62	ex 412	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → (𝑢 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} → 𝑢 ∈ 𝑡))
64	63	ssrdv 3989	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ⊆ 𝑡)
65	64	sspwd 4616	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ⊆ 𝒫 𝑡)
66		simp-4r 781	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
67	65, 66	sseldd 3984	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → 𝑥 ∈ 𝒫 𝑡)
68		simpllr 773	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦))
69	15	simp3d 1143	. . . . . . . . . . . . 13 ⊢ (𝑡 ∈ 𝐿 → ∀𝑥 ∈ 𝒫 𝑡((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
70	69	r19.21bi 3247	. . . . . . . . . . . 12 ⊢ ((𝑡 ∈ 𝐿 ∧ 𝑥 ∈ 𝒫 𝑡) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ 𝑡))
71	70	imp 406	. . . . . . . . . . 11 ⊢ (((𝑡 ∈ 𝐿 ∧ 𝑥 ∈ 𝒫 𝑡) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ 𝑡)
72	53, 67, 68, 71	syl21anc 835	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) ∧ 𝐴 ⊆ 𝑡) → ∪ 𝑥 ∈ 𝑡)
73	72	ex 412	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) ∧ 𝑡 ∈ 𝐿) → (𝐴 ⊆ 𝑡 → ∪ 𝑥 ∈ 𝑡))
74	73	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → (𝑡 ∈ 𝐿 → (𝐴 ⊆ 𝑡 → ∪ 𝑥 ∈ 𝑡)))
75	52, 74	ralrimi 3253	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → ∪ 𝑥 ∈ 𝑡))
76		vuniex 7732	. . . . . . . 8 ⊢ ∪ 𝑥 ∈ V
77	76	elintrab 4965	. . . . . . 7 ⊢ (∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ↔ ∀𝑡 ∈ 𝐿 (𝐴 ⊆ 𝑡 → ∪ 𝑥 ∈ 𝑡))
78	75, 77	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) ∧ (𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦)) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})
79	78	ex 412	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}) → ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}))
80	79	ralrimiva 3145	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}))
81	22, 47, 80	3jca 1127	. . 3 ⊢ (𝜑 → (∅ ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} (𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡})))
82	13, 81	jca 511	. 2 ⊢ (𝜑 → (∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} (𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}))))
83	4	isldsys 33449	. 2 ⊢ (∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝐿 ↔ (∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝒫 𝒫 𝑂 ∧ (∅ ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} (𝑂 ∖ 𝑥) ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∧ ∀𝑥 ∈ 𝒫 ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ((𝑥 ≼ ω ∧ Disj 𝑦 ∈ 𝑥 𝑦) → ∪ 𝑥 ∈ ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡}))))
84	82, 83	sylibr 233	1 ⊢ (𝜑 → ∩ {𝑡 ∈ 𝐿 ∣ 𝐴 ⊆ 𝑡} ∈ 𝐿)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 Vcvv 3473 ∖ cdif 3946 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∩ cint 4951 Disj wdisj 5114 class class class wbr 5149 ‘cfv 6544 ωcom 7858 ≼ cdom 8940 sigAlgebracsiga 33401

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-inf2 9639 ax-ac2 10461

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-oi 9508 df-dju 9899 df-card 9937 df-acn 9940 df-ac 10114 df-siga 33402

This theorem is referenced by: ldgenpisys 33459 dynkin 33460

W3C validator