Theorem tfinnn 4535

Description: T-raising of a set of naturals. Theorem X.1.46 of [Rosser] p. 532. (Contributed by SF, 30-Jan-2015.)

Assertion

Ref	Expression
tfinnn	⊢ ((N ∈ Nn ∧ A ⊆ Nn ∧ A ∈ N) → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)

Distinct variable groups:   N,a,x   A,a,x

Proof of Theorem tfinnn

Dummy variables y n b k w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tfinnnlem1 4534	. . . . 5 ⊢ {n ∣ ∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n)} ∈ V
2		tfineq 4489	. . . . . . . . . 10 ⊢ (n = 0c → Tfin n = Tfin 0c)
3		tfin0c 4498	. . . . . . . . . 10 ⊢ Tfin 0c = 0c
4	2, 3	syl6eq 2401	. . . . . . . . 9 ⊢ (n = 0c → Tfin n = 0c)
5	4	eleq2d 2420	. . . . . . . 8 ⊢ (n = 0c → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n ↔ {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c))
6	5	imbi2d 307	. . . . . . 7 ⊢ (n = 0c → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c)))
7	6	raleqbi1dv 2816	. . . . . 6 ⊢ (n = 0c → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ ∀y ∈ 0c (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c)))
8		df-ral 2620	. . . . . . 7 ⊢ (∀y ∈ 0c (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c) ↔ ∀y(y ∈ 0c → (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c)))
9		el0c 4422	. . . . . . . . 9 ⊢ (y ∈ 0c ↔ y = ∅)
10		el0c 4422	. . . . . . . . . . 11 ⊢ ({a ∣ ∃x ∈ y a = Tfin x} ∈ 0c ↔ {a ∣ ∃x ∈ y a = Tfin x} = ∅)
11		ab0 3570	. . . . . . . . . . 11 ⊢ ({a ∣ ∃x ∈ y a = Tfin x} = ∅ ↔ ∀a ¬ ∃x ∈ y a = Tfin x)
12	10, 11	bitri 240	. . . . . . . . . 10 ⊢ ({a ∣ ∃x ∈ y a = Tfin x} ∈ 0c ↔ ∀a ¬ ∃x ∈ y a = Tfin x)
13	12	imbi2i 303	. . . . . . . . 9 ⊢ ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c) ↔ (y ⊆ Nn → ∀a ¬ ∃x ∈ y a = Tfin x))
14	9, 13	imbi12i 316	. . . . . . . 8 ⊢ ((y ∈ 0c → (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c)) ↔ (y = ∅ → (y ⊆ Nn → ∀a ¬ ∃x ∈ y a = Tfin x)))
15	14	albii 1566	. . . . . . 7 ⊢ (∀y(y ∈ 0c → (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c)) ↔ ∀y(y = ∅ → (y ⊆ Nn → ∀a ¬ ∃x ∈ y a = Tfin x)))
16		0ex 4111	. . . . . . . 8 ⊢ ∅ ∈ V
17		sseq1 3293	. . . . . . . . 9 ⊢ (y = ∅ → (y ⊆ Nn ↔ ∅ ⊆ Nn ))
18		rexeq 2809	. . . . . . . . . . 11 ⊢ (y = ∅ → (∃x ∈ y a = Tfin x ↔ ∃x ∈ ∅ a = Tfin x))
19	18	notbid 285	. . . . . . . . . 10 ⊢ (y = ∅ → (¬ ∃x ∈ y a = Tfin x ↔ ¬ ∃x ∈ ∅ a = Tfin x))
20	19	albidv 1625	. . . . . . . . 9 ⊢ (y = ∅ → (∀a ¬ ∃x ∈ y a = Tfin x ↔ ∀a ¬ ∃x ∈ ∅ a = Tfin x))
21	17, 20	imbi12d 311	. . . . . . . 8 ⊢ (y = ∅ → ((y ⊆ Nn → ∀a ¬ ∃x ∈ y a = Tfin x) ↔ (∅ ⊆ Nn → ∀a ¬ ∃x ∈ ∅ a = Tfin x)))
22	16, 21	ceqsalv 2886	. . . . . . 7 ⊢ (∀y(y = ∅ → (y ⊆ Nn → ∀a ¬ ∃x ∈ y a = Tfin x)) ↔ (∅ ⊆ Nn → ∀a ¬ ∃x ∈ ∅ a = Tfin x))
23	8, 15, 22	3bitri 262	. . . . . 6 ⊢ (∀y ∈ 0c (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ 0c) ↔ (∅ ⊆ Nn → ∀a ¬ ∃x ∈ ∅ a = Tfin x))
24	7, 23	syl6bb 252	. . . . 5 ⊢ (n = 0c → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ (∅ ⊆ Nn → ∀a ¬ ∃x ∈ ∅ a = Tfin x)))
25		tfineq 4489	. . . . . . . 8 ⊢ (n = k → Tfin n = Tfin k)
26	25	eleq2d 2420	. . . . . . 7 ⊢ (n = k → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n ↔ {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k))
27	26	imbi2d 307	. . . . . 6 ⊢ (n = k → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)))
28	27	raleqbi1dv 2816	. . . . 5 ⊢ (n = k → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)))
29		tfineq 4489	. . . . . . . . 9 ⊢ (n = (k +c 1c) → Tfin n = Tfin (k +c 1c))
30	29	eleq2d 2420	. . . . . . . 8 ⊢ (n = (k +c 1c) → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n ↔ {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c)))
31	30	imbi2d 307	. . . . . . 7 ⊢ (n = (k +c 1c) → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c))))
32	31	raleqbi1dv 2816	. . . . . 6 ⊢ (n = (k +c 1c) → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ ∀y ∈ (k +c 1c)(y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c))))
33		sseq1 3293	. . . . . . . 8 ⊢ (y = z → (y ⊆ Nn ↔ z ⊆ Nn ))
34		rexeq 2809	. . . . . . . . . 10 ⊢ (y = z → (∃x ∈ y a = Tfin x ↔ ∃x ∈ z a = Tfin x))
35	34	abbidv 2468	. . . . . . . . 9 ⊢ (y = z → {a ∣ ∃x ∈ y a = Tfin x} = {a ∣ ∃x ∈ z a = Tfin x})
36	35	eleq1d 2419	. . . . . . . 8 ⊢ (y = z → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c) ↔ {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c)))
37	33, 36	imbi12d 311	. . . . . . 7 ⊢ (y = z → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c)) ↔ (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c))))
38	37	cbvralv 2836	. . . . . 6 ⊢ (∀y ∈ (k +c 1c)(y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin (k +c 1c)) ↔ ∀z ∈ (k +c 1c)(z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c)))
39	32, 38	syl6bb 252	. . . . 5 ⊢ (n = (k +c 1c) → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ ∀z ∈ (k +c 1c)(z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c))))
40		tfineq 4489	. . . . . . . 8 ⊢ (n = N → Tfin n = Tfin N)
41	40	eleq2d 2420	. . . . . . 7 ⊢ (n = N → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n ↔ {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N))
42	41	imbi2d 307	. . . . . 6 ⊢ (n = N → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N)))
43	42	raleqbi1dv 2816	. . . . 5 ⊢ (n = N → (∀y ∈ n (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin n) ↔ ∀y ∈ N (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N)))
44		rex0 3564	. . . . . . 7 ⊢ ¬ ∃x ∈ ∅ a = Tfin x
45	44	ax-gen 1546	. . . . . 6 ⊢ ∀a ¬ ∃x ∈ ∅ a = Tfin x
46	45	a1i 10	. . . . 5 ⊢ (∅ ⊆ Nn → ∀a ¬ ∃x ∈ ∅ a = Tfin x)
47		elsuc 4414	. . . . . . . . . . 11 ⊢ (z ∈ (k +c 1c) ↔ ∃b ∈ k ∃w ∈ ∼ bz = (b ∪ {w}))
48		sseq1 3293	. . . . . . . . . . . . . . . . . 18 ⊢ (y = b → (y ⊆ Nn ↔ b ⊆ Nn ))
49		rexeq 2809	. . . . . . . . . . . . . . . . . . . 20 ⊢ (y = b → (∃x ∈ y a = Tfin x ↔ ∃x ∈ b a = Tfin x))
50	49	abbidv 2468	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = b → {a ∣ ∃x ∈ y a = Tfin x} = {a ∣ ∃x ∈ b a = Tfin x})
51	50	eleq1d 2419	. . . . . . . . . . . . . . . . . 18 ⊢ (y = b → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k ↔ {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k))
52	48, 51	imbi12d 311	. . . . . . . . . . . . . . . . 17 ⊢ (y = b → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k) ↔ (b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k)))
53	52	rspcv 2952	. . . . . . . . . . . . . . . 16 ⊢ (b ∈ k → (∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k) → (b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k)))
54	53	ad2antrl 708	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) → (∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k) → (b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k)))
55		simprl 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → b ⊆ Nn )
56		simp3 957	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn ) ∧ {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k)
57		simplrr 737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → w ∈ ∼ b)
58		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ w ∈ V
59	58	elcompl 3226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (w ∈ ∼ b ↔ ¬ w ∈ b)
60	57, 59	sylib 188	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → ¬ w ∈ b)
61		elequ1 1713	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (w = x → (w ∈ b ↔ x ∈ b))
62	61	notbid 285	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (w = x → (¬ w ∈ b ↔ ¬ x ∈ b))
63	60, 62	syl5ibcom 211	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → (w = x → ¬ x ∈ b))
64	63	con2d 107	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → (x ∈ b → ¬ w = x))
65	64	imp 418	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) → ¬ w = x)
66		simpll 730	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )))
67		simprr 733	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → w ∈ Nn )
68	66, 67	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → w ∈ Nn )
69	66, 55	syl 15	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → b ⊆ Nn )
70		simplr 731	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → x ∈ b)
71	69, 70	sseldd 3275	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → x ∈ Nn )
72		simpr 447	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → Tfin w = Tfin x)
73		tfin11 4494	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((w ∈ Nn ∧ x ∈ Nn ∧ Tfin w = Tfin x) → w = x)
74	68, 71, 72, 73	syl3anc 1182	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) ∧ Tfin w = Tfin x) → w = x)
75	65, 74	mtand 640	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) ∧ x ∈ b) → ¬ Tfin w = Tfin x)
76	75	nrexdv 2718	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → ¬ ∃x ∈ b Tfin w = Tfin x)
77	76	3adant3 975	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn ) ∧ {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ¬ ∃x ∈ b Tfin w = Tfin x)
78		tfinex 4486	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Tfin w ∈ V
79		eqeq1 2359	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (a = Tfin w → (a = Tfin x ↔ Tfin w = Tfin x))
80	79	rexbidv 2636	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (a = Tfin w → (∃x ∈ b a = Tfin x ↔ ∃x ∈ b Tfin w = Tfin x))
81	78, 80	elab 2986	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( Tfin w ∈ {a ∣ ∃x ∈ b a = Tfin x} ↔ ∃x ∈ b Tfin w = Tfin x)
82	77, 81	sylnibr 296	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn ) ∧ {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ¬ Tfin w ∈ {a ∣ ∃x ∈ b a = Tfin x})
83	78	elsuci 4415	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (({a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k ∧ ¬ Tfin w ∈ {a ∣ ∃x ∈ b a = Tfin x}) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c))
84	56, 82, 83	syl2anc 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn ) ∧ {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c))
85	84	3expia 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → ({a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c)))
86	55, 85	embantd 50	. . . . . . . . . . . . . . . . . 18 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ (b ⊆ Nn ∧ w ∈ Nn )) → ((b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c)))
87	86	ex 423	. . . . . . . . . . . . . . . . 17 ⊢ ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) → ((b ⊆ Nn ∧ w ∈ Nn ) → ((b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c))))
88	87	com23 72	. . . . . . . . . . . . . . . 16 ⊢ ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) → ((b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → ((b ⊆ Nn ∧ w ∈ Nn ) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c))))
89		sseq1 3293	. . . . . . . . . . . . . . . . . . 19 ⊢ (z = (b ∪ {w}) → (z ⊆ Nn ↔ (b ∪ {w}) ⊆ Nn ))
90	58	snss 3839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (w ∈ Nn ↔ {w} ⊆ Nn )
91	90	anbi2i 675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((b ⊆ Nn ∧ w ∈ Nn ) ↔ (b ⊆ Nn ∧ {w} ⊆ Nn ))
92		unss 3438	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((b ⊆ Nn ∧ {w} ⊆ Nn ) ↔ (b ∪ {w}) ⊆ Nn )
93	91, 92	bitr2i 241	. . . . . . . . . . . . . . . . . . 19 ⊢ ((b ∪ {w}) ⊆ Nn ↔ (b ⊆ Nn ∧ w ∈ Nn ))
94	89, 93	syl6bb 252	. . . . . . . . . . . . . . . . . 18 ⊢ (z = (b ∪ {w}) → (z ⊆ Nn ↔ (b ⊆ Nn ∧ w ∈ Nn )))
95		rexeq 2809	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (z = (b ∪ {w}) → (∃x ∈ z a = Tfin x ↔ ∃x ∈ (b ∪ {w})a = Tfin x))
96		rexun 3444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∃x ∈ (b ∪ {w})a = Tfin x ↔ (∃x ∈ b a = Tfin x ∨ ∃x ∈ {w}a = Tfin x))
97	95, 96	syl6bb 252	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (z = (b ∪ {w}) → (∃x ∈ z a = Tfin x ↔ (∃x ∈ b a = Tfin x ∨ ∃x ∈ {w}a = Tfin x)))
98	97	abbidv 2468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (z = (b ∪ {w}) → {a ∣ ∃x ∈ z a = Tfin x} = {a ∣ (∃x ∈ b a = Tfin x ∨ ∃x ∈ {w}a = Tfin x)})
99		df-sn 3742	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ { Tfin w} = {a ∣ a = Tfin w}
100		tfineq 4489	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (x = w → Tfin x = Tfin w)
101	100	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (x = w → (a = Tfin x ↔ a = Tfin w))
102	58, 101	rexsn 3769	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∃x ∈ {w}a = Tfin x ↔ a = Tfin w)
103	102	abbii 2466	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {a ∣ ∃x ∈ {w}a = Tfin x} = {a ∣ a = Tfin w}
104	99, 103	eqtr4i 2376	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ { Tfin w} = {a ∣ ∃x ∈ {w}a = Tfin x}
105	104	uneq2i 3416	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) = ({a ∣ ∃x ∈ b a = Tfin x} ∪ {a ∣ ∃x ∈ {w}a = Tfin x})
106		unab 3522	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ({a ∣ ∃x ∈ b a = Tfin x} ∪ {a ∣ ∃x ∈ {w}a = Tfin x}) = {a ∣ (∃x ∈ b a = Tfin x ∨ ∃x ∈ {w}a = Tfin x)}
107	105, 106	eqtr2i 2374	. . . . . . . . . . . . . . . . . . . 20 ⊢ {a ∣ (∃x ∈ b a = Tfin x ∨ ∃x ∈ {w}a = Tfin x)} = ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w})
108	98, 107	syl6eq 2401	. . . . . . . . . . . . . . . . . . 19 ⊢ (z = (b ∪ {w}) → {a ∣ ∃x ∈ z a = Tfin x} = ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}))
109	108	eleq1d 2419	. . . . . . . . . . . . . . . . . 18 ⊢ (z = (b ∪ {w}) → ({a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c) ↔ ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c)))
110	94, 109	imbi12d 311	. . . . . . . . . . . . . . . . 17 ⊢ (z = (b ∪ {w}) → ((z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c)) ↔ ((b ⊆ Nn ∧ w ∈ Nn ) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c))))
111	110	biimprcd 216	. . . . . . . . . . . . . . . 16 ⊢ (((b ⊆ Nn ∧ w ∈ Nn ) → ({a ∣ ∃x ∈ b a = Tfin x} ∪ { Tfin w}) ∈ ( Tfin k +c 1c)) → (z = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))))
112	88, 111	syl6 29	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) → ((b ⊆ Nn → {a ∣ ∃x ∈ b a = Tfin x} ∈ Tfin k) → (z = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c)))))
113	54, 112	syld 40	. . . . . . . . . . . . . 14 ⊢ ((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) → (∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k) → (z = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c)))))
114	113	imp 418	. . . . . . . . . . . . 13 ⊢ (((k ∈ Nn ∧ (b ∈ k ∧ w ∈ ∼ b)) ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) → (z = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))))
115	114	an32s 779	. . . . . . . . . . . 12 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (b ∈ k ∧ w ∈ ∼ b)) → (z = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))))
116	115	rexlimdvva 2746	. . . . . . . . . . 11 ⊢ ((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) → (∃b ∈ k ∃w ∈ ∼ bz = (b ∪ {w}) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))))
117	47, 116	syl5bi 208	. . . . . . . . . 10 ⊢ ((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) → (z ∈ (k +c 1c) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))))
118	117	imp32 422	. . . . . . . . 9 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (z ∈ (k +c 1c) ∧ z ⊆ Nn )) → {a ∣ ∃x ∈ z a = Tfin x} ∈ ( Tfin k +c 1c))
119		simpll 730	. . . . . . . . . 10 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (z ∈ (k +c 1c) ∧ z ⊆ Nn )) → k ∈ Nn )
120		ne0i 3557	. . . . . . . . . . 11 ⊢ (z ∈ (k +c 1c) → (k +c 1c) ≠ ∅)
121	120	ad2antrl 708	. . . . . . . . . 10 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (z ∈ (k +c 1c) ∧ z ⊆ Nn )) → (k +c 1c) ≠ ∅)
122		tfinsuc 4499	. . . . . . . . . 10 ⊢ ((k ∈ Nn ∧ (k +c 1c) ≠ ∅) → Tfin (k +c 1c) = ( Tfin k +c 1c))
123	119, 121, 122	syl2anc 642	. . . . . . . . 9 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (z ∈ (k +c 1c) ∧ z ⊆ Nn )) → Tfin (k +c 1c) = ( Tfin k +c 1c))
124	118, 123	eleqtrrd 2430	. . . . . . . 8 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ (z ∈ (k +c 1c) ∧ z ⊆ Nn )) → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c))
125	124	expr 598	. . . . . . 7 ⊢ (((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) ∧ z ∈ (k +c 1c)) → (z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c)))
126	125	ralrimiva 2698	. . . . . 6 ⊢ ((k ∈ Nn ∧ ∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k)) → ∀z ∈ (k +c 1c)(z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c)))
127	126	ex 423	. . . . 5 ⊢ (k ∈ Nn → (∀y ∈ k (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin k) → ∀z ∈ (k +c 1c)(z ⊆ Nn → {a ∣ ∃x ∈ z a = Tfin x} ∈ Tfin (k +c 1c))))
128	1, 24, 28, 39, 43, 46, 127	finds 4412	. . . 4 ⊢ (N ∈ Nn → ∀y ∈ N (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N))
129		sseq1 3293	. . . . . 6 ⊢ (y = A → (y ⊆ Nn ↔ A ⊆ Nn ))
130		rexeq 2809	. . . . . . . 8 ⊢ (y = A → (∃x ∈ y a = Tfin x ↔ ∃x ∈ A a = Tfin x))
131	130	abbidv 2468	. . . . . . 7 ⊢ (y = A → {a ∣ ∃x ∈ y a = Tfin x} = {a ∣ ∃x ∈ A a = Tfin x})
132	131	eleq1d 2419	. . . . . 6 ⊢ (y = A → ({a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N ↔ {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N))
133	129, 132	imbi12d 311	. . . . 5 ⊢ (y = A → ((y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N) ↔ (A ⊆ Nn → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)))
134	133	rspccv 2953	. . . 4 ⊢ (∀y ∈ N (y ⊆ Nn → {a ∣ ∃x ∈ y a = Tfin x} ∈ Tfin N) → (A ∈ N → (A ⊆ Nn → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)))
135	128, 134	syl 15	. . 3 ⊢ (N ∈ Nn → (A ∈ N → (A ⊆ Nn → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)))
136	135	com23 72	. 2 ⊢ (N ∈ Nn → (A ⊆ Nn → (A ∈ N → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)))
137	136	3imp 1145	1 ⊢ ((N ∈ Nn ∧ A ⊆ Nn ∧ A ∈ N) → {a ∣ ∃x ∈ A a = Tfin x} ∈ Tfin N)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 357   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339   ≠ wne 2517  ∀wral 2615  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208   ⊆ wss 3258  ∅c0 3551  {csn 3738  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376   T_fin ctfin 4436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-tfin 4444

This theorem is referenced by:  vfinncsp  4555