Theorem nnadjoin 4521

Description: Adjoining a new element to every member of L does not change its size. Theorem X.1.39 of [Rosser] p. 530. (Contributed by SF, 29-Jan-2015.)

Assertion

Ref	Expression
nnadjoin	⊢ ((N ∈ Nn ∧ L ∈ N ∧ X ∈ ∼ ∪L) → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N)

Distinct variable groups:   L,b,x   X,b,x

Allowed substitution hints:   N(x,b)

Proof of Theorem nnadjoin

Dummy variables l y n a c k z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3745	. . . . . . . . . . 11 ⊢ (y = X → {y} = {X})
2	1	uneq2d 3419	. . . . . . . . . 10 ⊢ (y = X → (b ∪ {y}) = (b ∪ {X}))
3	2	eqeq2d 2364	. . . . . . . . 9 ⊢ (y = X → (x = (b ∪ {y}) ↔ x = (b ∪ {X})))
4	3	rexbidv 2636	. . . . . . . 8 ⊢ (y = X → (∃b ∈ L x = (b ∪ {y}) ↔ ∃b ∈ L x = (b ∪ {X})))
5	4	abbidv 2468	. . . . . . 7 ⊢ (y = X → {x ∣ ∃b ∈ L x = (b ∪ {y})} = {x ∣ ∃b ∈ L x = (b ∪ {X})})
6	5	eleq1d 2419	. . . . . 6 ⊢ (y = X → ({x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N ↔ {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N))
7	6	imbi2d 307	. . . . 5 ⊢ (y = X → ((L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N) ↔ (L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N)))
8	7	imbi2d 307	. . . 4 ⊢ (y = X → ((N ∈ Nn → (L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N)) ↔ (N ∈ Nn → (L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N))))
9		nnadjoinlem1 4520	. . . . . . 7 ⊢ {n ∣ ∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n)} ∈ V
10		eleq2 2414	. . . . . . . . . . 11 ⊢ (n = 0c → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n ↔ {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ 0c))
11		el0c 4422	. . . . . . . . . . . 12 ⊢ ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ 0c ↔ {x ∣ ∃b ∈ l x = (b ∪ {y})} = ∅)
12		ab0 3570	. . . . . . . . . . . 12 ⊢ ({x ∣ ∃b ∈ l x = (b ∪ {y})} = ∅ ↔ ∀x ¬ ∃b ∈ l x = (b ∪ {y}))
13	11, 12	bitri 240	. . . . . . . . . . 11 ⊢ ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ 0c ↔ ∀x ¬ ∃b ∈ l x = (b ∪ {y}))
14	10, 13	syl6bb 252	. . . . . . . . . 10 ⊢ (n = 0c → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n ↔ ∀x ¬ ∃b ∈ l x = (b ∪ {y})))
15	14	imbi2d 307	. . . . . . . . 9 ⊢ (n = 0c → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))))
16	15	raleqbi1dv 2816	. . . . . . . 8 ⊢ (n = 0c → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ ∀l ∈ 0c (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))))
17		df-ral 2620	. . . . . . . . 9 ⊢ (∀l ∈ 0c (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y})) ↔ ∀l(l ∈ 0c → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))))
18		el0c 4422	. . . . . . . . . . 11 ⊢ (l ∈ 0c ↔ l = ∅)
19	18	imbi1i 315	. . . . . . . . . 10 ⊢ ((l ∈ 0c → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))) ↔ (l = ∅ → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))))
20	19	albii 1566	. . . . . . . . 9 ⊢ (∀l(l ∈ 0c → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))) ↔ ∀l(l = ∅ → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))))
21		0ex 4111	. . . . . . . . . 10 ⊢ ∅ ∈ V
22		unieq 3901	. . . . . . . . . . . . 13 ⊢ (l = ∅ → ∪l = ∪∅)
23	22	compleqd 3246	. . . . . . . . . . . 12 ⊢ (l = ∅ → ∼ ∪l = ∼ ∪∅)
24	23	eleq2d 2420	. . . . . . . . . . 11 ⊢ (l = ∅ → (y ∈ ∼ ∪l ↔ y ∈ ∼ ∪∅))
25		rexeq 2809	. . . . . . . . . . . . 13 ⊢ (l = ∅ → (∃b ∈ l x = (b ∪ {y}) ↔ ∃b ∈ ∅ x = (b ∪ {y})))
26	25	notbid 285	. . . . . . . . . . . 12 ⊢ (l = ∅ → (¬ ∃b ∈ l x = (b ∪ {y}) ↔ ¬ ∃b ∈ ∅ x = (b ∪ {y})))
27	26	albidv 1625	. . . . . . . . . . 11 ⊢ (l = ∅ → (∀x ¬ ∃b ∈ l x = (b ∪ {y}) ↔ ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y})))
28	24, 27	imbi12d 311	. . . . . . . . . 10 ⊢ (l = ∅ → ((y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y})) ↔ (y ∈ ∼ ∪∅ → ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y}))))
29	21, 28	ceqsalv 2886	. . . . . . . . 9 ⊢ (∀l(l = ∅ → (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y}))) ↔ (y ∈ ∼ ∪∅ → ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y})))
30	17, 20, 29	3bitrri 263	. . . . . . . 8 ⊢ ((y ∈ ∼ ∪∅ → ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y})) ↔ ∀l ∈ 0c (y ∈ ∼ ∪l → ∀x ¬ ∃b ∈ l x = (b ∪ {y})))
31	16, 30	syl6bbr 254	. . . . . . 7 ⊢ (n = 0c → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ (y ∈ ∼ ∪∅ → ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y}))))
32		eleq2 2414	. . . . . . . . 9 ⊢ (n = k → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n ↔ {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k))
33	32	imbi2d 307	. . . . . . . 8 ⊢ (n = k → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)))
34	33	raleqbi1dv 2816	. . . . . . 7 ⊢ (n = k → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)))
35		eleq2 2414	. . . . . . . . . 10 ⊢ (n = (k +c 1c) → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n ↔ {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c)))
36	35	imbi2d 307	. . . . . . . . 9 ⊢ (n = (k +c 1c) → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c))))
37	36	raleqbi1dv 2816	. . . . . . . 8 ⊢ (n = (k +c 1c) → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ ∀l ∈ (k +c 1c)(y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c))))
38		unieq 3901	. . . . . . . . . . . 12 ⊢ (l = a → ∪l = ∪a)
39	38	compleqd 3246	. . . . . . . . . . 11 ⊢ (l = a → ∼ ∪l = ∼ ∪a)
40	39	eleq2d 2420	. . . . . . . . . 10 ⊢ (l = a → (y ∈ ∼ ∪l ↔ y ∈ ∼ ∪a))
41		rexeq 2809	. . . . . . . . . . . 12 ⊢ (l = a → (∃b ∈ l x = (b ∪ {y}) ↔ ∃b ∈ a x = (b ∪ {y})))
42	41	abbidv 2468	. . . . . . . . . . 11 ⊢ (l = a → {x ∣ ∃b ∈ l x = (b ∪ {y})} = {x ∣ ∃b ∈ a x = (b ∪ {y})})
43	42	eleq1d 2419	. . . . . . . . . 10 ⊢ (l = a → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c) ↔ {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c)))
44	40, 43	imbi12d 311	. . . . . . . . 9 ⊢ (l = a → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c)) ↔ (y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
45	44	cbvralv 2836	. . . . . . . 8 ⊢ (∀l ∈ (k +c 1c)(y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ (k +c 1c)) ↔ ∀a ∈ (k +c 1c)(y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c)))
46	37, 45	syl6bb 252	. . . . . . 7 ⊢ (n = (k +c 1c) → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ ∀a ∈ (k +c 1c)(y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
47		eleq2 2414	. . . . . . . . 9 ⊢ (n = N → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n ↔ {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N))
48	47	imbi2d 307	. . . . . . . 8 ⊢ (n = N → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N)))
49	48	raleqbi1dv 2816	. . . . . . 7 ⊢ (n = N → (∀l ∈ n (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ n) ↔ ∀l ∈ N (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N)))
50		rex0 3564	. . . . . . . . 9 ⊢ ¬ ∃b ∈ ∅ x = (b ∪ {y})
51	50	ax-gen 1546	. . . . . . . 8 ⊢ ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y})
52	51	a1i 10	. . . . . . 7 ⊢ (y ∈ ∼ ∪∅ → ∀x ¬ ∃b ∈ ∅ x = (b ∪ {y}))
53		elsuc 4414	. . . . . . . . . 10 ⊢ (a ∈ (k +c 1c) ↔ ∃c ∈ k ∃z ∈ ∼ ca = (c ∪ {z}))
54		unieq 3901	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (l = c → ∪l = ∪c)
55	54	compleqd 3246	. . . . . . . . . . . . . . . . . . . 20 ⊢ (l = c → ∼ ∪l = ∼ ∪c)
56	55	eleq2d 2420	. . . . . . . . . . . . . . . . . . 19 ⊢ (l = c → (y ∈ ∼ ∪l ↔ y ∈ ∼ ∪c))
57		rexeq 2809	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (l = c → (∃b ∈ l x = (b ∪ {y}) ↔ ∃b ∈ c x = (b ∪ {y})))
58	57	abbidv 2468	. . . . . . . . . . . . . . . . . . . 20 ⊢ (l = c → {x ∣ ∃b ∈ l x = (b ∪ {y})} = {x ∣ ∃b ∈ c x = (b ∪ {y})})
59	58	eleq1d 2419	. . . . . . . . . . . . . . . . . . 19 ⊢ (l = c → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k ↔ {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k))
60	56, 59	imbi12d 311	. . . . . . . . . . . . . . . . . 18 ⊢ (l = c → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) ↔ (y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k)))
61	60	rspcv 2952	. . . . . . . . . . . . . . . . 17 ⊢ (c ∈ k → (∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) → (y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k)))
62	61	adantr 451	. . . . . . . . . . . . . . . 16 ⊢ ((c ∈ k ∧ z ∈ ∼ c) → (∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) → (y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k)))
63	62	adantl 452	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) → (∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) → (y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k)))
64		elin 3220	. . . . . . . . . . . . . . . . 17 ⊢ (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) ↔ (y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z}))
65		simp3l 983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z})) → y ∈ ∼ ∪c)
66		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ z ∈ V
67	66	unisn 3908	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ∪{z} = z
68	67	compleqi 3245	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∼ ∪{z} = ∼ z
69	68	eleq2i 2417	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (y ∈ ∼ ∪{z} ↔ y ∈ ∼ z)
70	69	anbi2i 675	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z}) ↔ (y ∈ ∼ ∪c ∧ y ∈ ∼ z))
71		simpr 447	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k)
72		simpl2r 1009	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → z ∈ ∼ c)
73	66	elcompl 3226	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (z ∈ ∼ c ↔ ¬ z ∈ c)
74	72, 73	sylib 188	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ¬ z ∈ c)
75		eleq1a 2422	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (b ∈ c → (z = b → z ∈ c))
76	75	adantl 452	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → (z = b → z ∈ c))
77	74, 76	mtod 168	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ¬ z = b)
78		simpl3r 1011	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → y ∈ ∼ z)
79		vex 2863	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ y ∈ V
80	79	elcompl 3226	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (y ∈ ∼ z ↔ ¬ y ∈ z)
81	78, 80	sylib 188	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ¬ y ∈ z)
82		simp3l 983	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) → y ∈ ∼ ∪c)
83	79	elcompl 3226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (y ∈ ∼ ∪c ↔ ¬ y ∈ ∪c)
84	82, 83	sylib 188	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) → ¬ y ∈ ∪c)
85		elunii 3897	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((y ∈ b ∧ b ∈ c) → y ∈ ∪c)
86	85	expcom 424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (b ∈ c → (y ∈ b → y ∈ ∪c))
87	86	con3d 125	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (b ∈ c → (¬ y ∈ ∪c → ¬ y ∈ b))
88	84, 87	mpan9 455	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ¬ y ∈ b)
89		adj11 3890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((¬ y ∈ z ∧ ¬ y ∈ b) → ((z ∪ {y}) = (b ∪ {y}) ↔ z = b))
90	81, 88, 89	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ((z ∪ {y}) = (b ∪ {y}) ↔ z = b))
91	77, 90	mtbird 292	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ b ∈ c) → ¬ (z ∪ {y}) = (b ∪ {y}))
92	91	nrexdv 2718	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) → ¬ ∃b ∈ c (z ∪ {y}) = (b ∪ {y}))
93		eqeq1 2359	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (x = (z ∪ {y}) → (x = (b ∪ {y}) ↔ (z ∪ {y}) = (b ∪ {y})))
94	93	rexbidv 2636	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (x = (z ∪ {y}) → (∃b ∈ c x = (b ∪ {y}) ↔ ∃b ∈ c (z ∪ {y}) = (b ∪ {y})))
95	94	elabg 2987	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})} → ((z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})} ↔ ∃b ∈ c (z ∪ {y}) = (b ∪ {y})))
96	95	ibi 232	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})} → ∃b ∈ c (z ∪ {y}) = (b ∪ {y}))
97	92, 96	nsyl 113	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) → ¬ (z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})})
98	97	adantr 451	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → ¬ (z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})})
99		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {y} ∈ V
100	66, 99	unex 4107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (z ∪ {y}) ∈ V
101	100	elsuci 4415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (({x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k ∧ ¬ (z ∪ {y}) ∈ {x ∣ ∃b ∈ c x = (b ∪ {y})}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))
102	71, 98, 101	syl2anc 642	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) ∧ {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))
103	102	ex 423	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ z)) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
104	70, 103	syl3an3b 1220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z})) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
105	65, 104	embantd 50	. . . . . . . . . . . . . . . . . 18 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c) ∧ (y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z})) → ((y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
106	105	3expia 1153	. . . . . . . . . . . . . . . . 17 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) → ((y ∈ ∼ ∪c ∧ y ∈ ∼ ∪{z}) → ((y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))))
107	64, 106	syl5bi 208	. . . . . . . . . . . . . . . 16 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) → (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ((y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))))
108	107	com23 72	. . . . . . . . . . . . . . 15 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) → ((y ∈ ∼ ∪c → {x ∣ ∃b ∈ c x = (b ∪ {y})} ∈ k) → (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))))
109	63, 108	syld 40	. . . . . . . . . . . . . 14 ⊢ ((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) → (∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) → (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))))
110	109	imp 418	. . . . . . . . . . . . 13 ⊢ (((k ∈ Nn ∧ (c ∈ k ∧ z ∈ ∼ c)) ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) → (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
111	110	an32s 779	. . . . . . . . . . . 12 ⊢ (((k ∈ Nn ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) ∧ (c ∈ k ∧ z ∈ ∼ c)) → (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
112		unieq 3901	. . . . . . . . . . . . . . . 16 ⊢ (a = (c ∪ {z}) → ∪a = ∪(c ∪ {z}))
113	112	compleqd 3246	. . . . . . . . . . . . . . 15 ⊢ (a = (c ∪ {z}) → ∼ ∪a = ∼ ∪(c ∪ {z}))
114		uniun 3911	. . . . . . . . . . . . . . . . 17 ⊢ ∪(c ∪ {z}) = (∪c ∪ ∪{z})
115	114	compleqi 3245	. . . . . . . . . . . . . . . 16 ⊢ ∼ ∪(c ∪ {z}) = ∼ (∪c ∪ ∪{z})
116		iunin 3548	. . . . . . . . . . . . . . . 16 ⊢ ∼ (∪c ∪ ∪{z}) = ( ∼ ∪c ∩ ∼ ∪{z})
117	115, 116	eqtri 2373	. . . . . . . . . . . . . . 15 ⊢ ∼ ∪(c ∪ {z}) = ( ∼ ∪c ∩ ∼ ∪{z})
118	113, 117	syl6eq 2401	. . . . . . . . . . . . . 14 ⊢ (a = (c ∪ {z}) → ∼ ∪a = ( ∼ ∪c ∩ ∼ ∪{z}))
119	118	eleq2d 2420	. . . . . . . . . . . . 13 ⊢ (a = (c ∪ {z}) → (y ∈ ∼ ∪a ↔ y ∈ ( ∼ ∪c ∩ ∼ ∪{z})))
120		rexeq 2809	. . . . . . . . . . . . . . . 16 ⊢ (a = (c ∪ {z}) → (∃b ∈ a x = (b ∪ {y}) ↔ ∃b ∈ (c ∪ {z})x = (b ∪ {y})))
121	120	abbidv 2468	. . . . . . . . . . . . . . 15 ⊢ (a = (c ∪ {z}) → {x ∣ ∃b ∈ a x = (b ∪ {y})} = {x ∣ ∃b ∈ (c ∪ {z})x = (b ∪ {y})})
122		unab 3522	. . . . . . . . . . . . . . . 16 ⊢ ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {x ∣ x = (z ∪ {y})}) = {x ∣ (∃b ∈ c x = (b ∪ {y}) ∨ x = (z ∪ {y}))}
123		df-sn 3742	. . . . . . . . . . . . . . . . 17 ⊢ {(z ∪ {y})} = {x ∣ x = (z ∪ {y})}
124	123	uneq2i 3416	. . . . . . . . . . . . . . . 16 ⊢ ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) = ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {x ∣ x = (z ∪ {y})})
125		rexun 3444	. . . . . . . . . . . . . . . . . 18 ⊢ (∃b ∈ (c ∪ {z})x = (b ∪ {y}) ↔ (∃b ∈ c x = (b ∪ {y}) ∨ ∃b ∈ {z}x = (b ∪ {y})))
126		uneq1 3412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (b = z → (b ∪ {y}) = (z ∪ {y}))
127	126	eqeq2d 2364	. . . . . . . . . . . . . . . . . . . 20 ⊢ (b = z → (x = (b ∪ {y}) ↔ x = (z ∪ {y})))
128	66, 127	rexsn 3769	. . . . . . . . . . . . . . . . . . 19 ⊢ (∃b ∈ {z}x = (b ∪ {y}) ↔ x = (z ∪ {y}))
129	128	orbi2i 505	. . . . . . . . . . . . . . . . . 18 ⊢ ((∃b ∈ c x = (b ∪ {y}) ∨ ∃b ∈ {z}x = (b ∪ {y})) ↔ (∃b ∈ c x = (b ∪ {y}) ∨ x = (z ∪ {y})))
130	125, 129	bitri 240	. . . . . . . . . . . . . . . . 17 ⊢ (∃b ∈ (c ∪ {z})x = (b ∪ {y}) ↔ (∃b ∈ c x = (b ∪ {y}) ∨ x = (z ∪ {y})))
131	130	abbii 2466	. . . . . . . . . . . . . . . 16 ⊢ {x ∣ ∃b ∈ (c ∪ {z})x = (b ∪ {y})} = {x ∣ (∃b ∈ c x = (b ∪ {y}) ∨ x = (z ∪ {y}))}
132	122, 124, 131	3eqtr4ri 2384	. . . . . . . . . . . . . . 15 ⊢ {x ∣ ∃b ∈ (c ∪ {z})x = (b ∪ {y})} = ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})})
133	121, 132	syl6eq 2401	. . . . . . . . . . . . . 14 ⊢ (a = (c ∪ {z}) → {x ∣ ∃b ∈ a x = (b ∪ {y})} = ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}))
134	133	eleq1d 2419	. . . . . . . . . . . . 13 ⊢ (a = (c ∪ {z}) → ({x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c) ↔ ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c)))
135	119, 134	imbi12d 311	. . . . . . . . . . . 12 ⊢ (a = (c ∪ {z}) → ((y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c)) ↔ (y ∈ ( ∼ ∪c ∩ ∼ ∪{z}) → ({x ∣ ∃b ∈ c x = (b ∪ {y})} ∪ {(z ∪ {y})}) ∈ (k +c 1c))))
136	111, 135	syl5ibrcom 213	. . . . . . . . . . 11 ⊢ (((k ∈ Nn ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) ∧ (c ∈ k ∧ z ∈ ∼ c)) → (a = (c ∪ {z}) → (y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
137	136	rexlimdvva 2746	. . . . . . . . . 10 ⊢ ((k ∈ Nn ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) → (∃c ∈ k ∃z ∈ ∼ ca = (c ∪ {z}) → (y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
138	53, 137	syl5bi 208	. . . . . . . . 9 ⊢ ((k ∈ Nn ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) → (a ∈ (k +c 1c) → (y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
139	138	ralrimiv 2697	. . . . . . . 8 ⊢ ((k ∈ Nn ∧ ∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k)) → ∀a ∈ (k +c 1c)(y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c)))
140	139	ex 423	. . . . . . 7 ⊢ (k ∈ Nn → (∀l ∈ k (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ k) → ∀a ∈ (k +c 1c)(y ∈ ∼ ∪a → {x ∣ ∃b ∈ a x = (b ∪ {y})} ∈ (k +c 1c))))
141	9, 31, 34, 46, 49, 52, 140	finds 4412	. . . . . 6 ⊢ (N ∈ Nn → ∀l ∈ N (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N))
142		unieq 3901	. . . . . . . . . 10 ⊢ (l = L → ∪l = ∪L)
143	142	compleqd 3246	. . . . . . . . 9 ⊢ (l = L → ∼ ∪l = ∼ ∪L)
144	143	eleq2d 2420	. . . . . . . 8 ⊢ (l = L → (y ∈ ∼ ∪l ↔ y ∈ ∼ ∪L))
145		rexeq 2809	. . . . . . . . . 10 ⊢ (l = L → (∃b ∈ l x = (b ∪ {y}) ↔ ∃b ∈ L x = (b ∪ {y})))
146	145	abbidv 2468	. . . . . . . . 9 ⊢ (l = L → {x ∣ ∃b ∈ l x = (b ∪ {y})} = {x ∣ ∃b ∈ L x = (b ∪ {y})})
147	146	eleq1d 2419	. . . . . . . 8 ⊢ (l = L → ({x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N ↔ {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N))
148	144, 147	imbi12d 311	. . . . . . 7 ⊢ (l = L → ((y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N) ↔ (y ∈ ∼ ∪L → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N)))
149	148	rspccv 2953	. . . . . 6 ⊢ (∀l ∈ N (y ∈ ∼ ∪l → {x ∣ ∃b ∈ l x = (b ∪ {y})} ∈ N) → (L ∈ N → (y ∈ ∼ ∪L → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N)))
150	141, 149	syl 15	. . . . 5 ⊢ (N ∈ Nn → (L ∈ N → (y ∈ ∼ ∪L → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N)))
151	150	com3r 73	. . . 4 ⊢ (y ∈ ∼ ∪L → (N ∈ Nn → (L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {y})} ∈ N)))
152	8, 151	vtoclga 2921	. . 3 ⊢ (X ∈ ∼ ∪L → (N ∈ Nn → (L ∈ N → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N)))
153	152	com3l 75	. 2 ⊢ (N ∈ Nn → (L ∈ N → (X ∈ ∼ ∪L → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N)))
154	153	3imp 1145	1 ⊢ ((N ∈ Nn ∧ L ∈ N ∧ X ∈ ∼ ∪L) → {x ∣ ∃b ∈ L x = (b ∪ {X})} ∈ N)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   ∧ w3a 934  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208   ∩ cin 3209  ∅c0 3551  {csn 3738  ∪cuni 3892  1_cc1c 4135   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

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-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-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  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-0c 4378  df-addc 4379  df-nnc 4380

This theorem is referenced by:  nnadjoinpw  4522