Theorem ncfinraise 4482

Description: If two sets are in a particular finite cardinal, then their unit power sets are in the same natural. Theorem X.1.25 of [Rosser] p. 527. (Contributed by SF, 21-Jan-2015.)

Assertion

Ref	Expression
ncfinraise	⊢ ((M ∈ Nn ∧ A ∈ M ∧ B ∈ M) → ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1B ∈ n))

Distinct variable groups:   A,n   B,n

Allowed substitution hint:   M(n)

Proof of Theorem ncfinraise

Dummy variables a b m c d k x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ncfinraiselem2 4481	. . . 4 ⊢ {m ∣ ∀a ∈ m ∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)} ∈ V
2		raleq 2808	. . . . 5 ⊢ (m = 0c → (∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀b ∈ 0c ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
3	2	raleqbi1dv 2816	. . . 4 ⊢ (m = 0c → (∀a ∈ m ∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀a ∈ 0c ∀b ∈ 0c ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
4		raleq 2808	. . . . 5 ⊢ (m = k → (∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
5	4	raleqbi1dv 2816	. . . 4 ⊢ (m = k → (∀a ∈ m ∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
6		raleq 2808	. . . . 5 ⊢ (m = (k +c 1c) → (∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀b ∈ (k +c 1c)∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
7	6	raleqbi1dv 2816	. . . 4 ⊢ (m = (k +c 1c) → (∀a ∈ m ∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀a ∈ (k +c 1c)∀b ∈ (k +c 1c)∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
8		raleq 2808	. . . . 5 ⊢ (m = M → (∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀b ∈ M ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
9	8	raleqbi1dv 2816	. . . 4 ⊢ (m = M → (∀a ∈ m ∀b ∈ m ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∀a ∈ M ∀b ∈ M ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
10		el0c 4422	. . . . . 6 ⊢ (a ∈ 0c ↔ a = ∅)
11		el0c 4422	. . . . . 6 ⊢ (b ∈ 0c ↔ b = ∅)
12		peano1 4403	. . . . . . . 8 ⊢ 0c ∈ Nn
13		nulel0c 4423	. . . . . . . . 9 ⊢ ∅ ∈ 0c
14	13, 13	pm3.2i 441	. . . . . . . 8 ⊢ (∅ ∈ 0c ∧ ∅ ∈ 0c)
15		eleq2 2414	. . . . . . . . . 10 ⊢ (n = 0c → (∅ ∈ n ↔ ∅ ∈ 0c))
16	15, 15	anbi12d 691	. . . . . . . . 9 ⊢ (n = 0c → ((∅ ∈ n ∧ ∅ ∈ n) ↔ (∅ ∈ 0c ∧ ∅ ∈ 0c)))
17	16	rspcev 2956	. . . . . . . 8 ⊢ ((0c ∈ Nn ∧ (∅ ∈ 0c ∧ ∅ ∈ 0c)) → ∃n ∈ Nn (∅ ∈ n ∧ ∅ ∈ n))
18	12, 14, 17	mp2an 653	. . . . . . 7 ⊢ ∃n ∈ Nn (∅ ∈ n ∧ ∅ ∈ n)
19		pw1eq 4144	. . . . . . . . . . 11 ⊢ (a = ∅ → 𝒫1a = 𝒫1∅)
20		pw10 4162	. . . . . . . . . . 11 ⊢ 𝒫1∅ = ∅
21	19, 20	syl6eq 2401	. . . . . . . . . 10 ⊢ (a = ∅ → 𝒫1a = ∅)
22	21	eleq1d 2419	. . . . . . . . 9 ⊢ (a = ∅ → (𝒫1a ∈ n ↔ ∅ ∈ n))
23		pw1eq 4144	. . . . . . . . . . 11 ⊢ (b = ∅ → 𝒫1b = 𝒫1∅)
24	23, 20	syl6eq 2401	. . . . . . . . . 10 ⊢ (b = ∅ → 𝒫1b = ∅)
25	24	eleq1d 2419	. . . . . . . . 9 ⊢ (b = ∅ → (𝒫1b ∈ n ↔ ∅ ∈ n))
26	22, 25	bi2anan9 843	. . . . . . . 8 ⊢ ((a = ∅ ∧ b = ∅) → ((𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ (∅ ∈ n ∧ ∅ ∈ n)))
27	26	rexbidv 2636	. . . . . . 7 ⊢ ((a = ∅ ∧ b = ∅) → (∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn (∅ ∈ n ∧ ∅ ∈ n)))
28	18, 27	mpbiri 224	. . . . . 6 ⊢ ((a = ∅ ∧ b = ∅) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
29	10, 11, 28	syl2anb 465	. . . . 5 ⊢ ((a ∈ 0c ∧ b ∈ 0c) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
30	29	rgen2a 2681	. . . 4 ⊢ ∀a ∈ 0c ∀b ∈ 0c ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)
31		nfv 1619	. . . . . . 7 ⊢ Ⅎa k ∈ Nn
32		nfra1 2665	. . . . . . 7 ⊢ Ⅎa∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)
33	31, 32	nfan 1824	. . . . . 6 ⊢ Ⅎa(k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
34		nfv 1619	. . . . . . . 8 ⊢ Ⅎb k ∈ Nn
35		nfra2 2669	. . . . . . . 8 ⊢ Ⅎb∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)
36	34, 35	nfan 1824	. . . . . . 7 ⊢ Ⅎb(k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
37		nfv 1619	. . . . . . 7 ⊢ Ⅎb a ∈ (k +c 1c)
38		reeanv 2779	. . . . . . . . . 10 ⊢ (∃c ∈ k ∃d ∈ k (∃x ∈ ∼ ca = (c ∪ {x}) ∧ ∃y ∈ ∼ db = (d ∪ {y})) ↔ (∃c ∈ k ∃x ∈ ∼ ca = (c ∪ {x}) ∧ ∃d ∈ k ∃y ∈ ∼ db = (d ∪ {y})))
39		reeanv 2779	. . . . . . . . . . 11 ⊢ (∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) ↔ (∃x ∈ ∼ ca = (c ∪ {x}) ∧ ∃y ∈ ∼ db = (d ∪ {y})))
40	39	2rexbii 2642	. . . . . . . . . 10 ⊢ (∃c ∈ k ∃d ∈ k ∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) ↔ ∃c ∈ k ∃d ∈ k (∃x ∈ ∼ ca = (c ∪ {x}) ∧ ∃y ∈ ∼ db = (d ∪ {y})))
41		elsuc 4414	. . . . . . . . . . 11 ⊢ (a ∈ (k +c 1c) ↔ ∃c ∈ k ∃x ∈ ∼ ca = (c ∪ {x}))
42		elsuc 4414	. . . . . . . . . . 11 ⊢ (b ∈ (k +c 1c) ↔ ∃d ∈ k ∃y ∈ ∼ db = (d ∪ {y}))
43	41, 42	anbi12i 678	. . . . . . . . . 10 ⊢ ((a ∈ (k +c 1c) ∧ b ∈ (k +c 1c)) ↔ (∃c ∈ k ∃x ∈ ∼ ca = (c ∪ {x}) ∧ ∃d ∈ k ∃y ∈ ∼ db = (d ∪ {y})))
44	38, 40, 43	3bitr4ri 269	. . . . . . . . 9 ⊢ ((a ∈ (k +c 1c) ∧ b ∈ (k +c 1c)) ↔ ∃c ∈ k ∃d ∈ k ∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})))
45		pw1eq 4144	. . . . . . . . . . . . . . . . 17 ⊢ (a = c → 𝒫1a = 𝒫1c)
46	45	eleq1d 2419	. . . . . . . . . . . . . . . 16 ⊢ (a = c → (𝒫1a ∈ n ↔ 𝒫1c ∈ n))
47	46	anbi1d 685	. . . . . . . . . . . . . . 15 ⊢ (a = c → ((𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ (𝒫1c ∈ n ∧ 𝒫1b ∈ n)))
48	47	rexbidv 2636	. . . . . . . . . . . . . 14 ⊢ (a = c → (∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1b ∈ n)))
49		pw1eq 4144	. . . . . . . . . . . . . . . . 17 ⊢ (b = d → 𝒫1b = 𝒫1d)
50	49	eleq1d 2419	. . . . . . . . . . . . . . . 16 ⊢ (b = d → (𝒫1b ∈ n ↔ 𝒫1d ∈ n))
51	50	anbi2d 684	. . . . . . . . . . . . . . 15 ⊢ (b = d → ((𝒫1c ∈ n ∧ 𝒫1b ∈ n) ↔ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)))
52	51	rexbidv 2636	. . . . . . . . . . . . . 14 ⊢ (b = d → (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)))
53	48, 52	rspc2v 2962	. . . . . . . . . . . . 13 ⊢ ((c ∈ k ∧ d ∈ k) → (∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) → ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)))
54	53	com12 27	. . . . . . . . . . . 12 ⊢ (∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) → ((c ∈ k ∧ d ∈ k) → ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)))
55		vex 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ x ∈ V
56	55	elcompl 3226	. . . . . . . . . . . . . . . . . . 19 ⊢ (x ∈ ∼ c ↔ ¬ x ∈ c)
57		vex 2863	. . . . . . . . . . . . . . . . . . . 20 ⊢ y ∈ V
58	57	elcompl 3226	. . . . . . . . . . . . . . . . . . 19 ⊢ (y ∈ ∼ d ↔ ¬ y ∈ d)
59	56, 58	anbi12i 678	. . . . . . . . . . . . . . . . . 18 ⊢ ((x ∈ ∼ c ∧ y ∈ ∼ d) ↔ (¬ x ∈ c ∧ ¬ y ∈ d))
60	59	anbi2i 675	. . . . . . . . . . . . . . . . 17 ⊢ (((c ∈ k ∧ d ∈ k) ∧ (x ∈ ∼ c ∧ y ∈ ∼ d)) ↔ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)))
61		peano2 4404	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (n ∈ Nn → (n +c 1c) ∈ Nn )
62	61	ad3antrrr 710	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → (n +c 1c) ∈ Nn )
63		simplrl 736	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) → 𝒫1c ∈ n)
64	63	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → 𝒫1c ∈ n)
65		simprrl 740	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ¬ x ∈ c)
66		snelpw1 4147	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({x} ∈ 𝒫1c ↔ x ∈ c)
67	65, 66	sylnibr 296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ¬ {x} ∈ 𝒫1c)
68		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {x} ∈ V
69	68	elsuci 4415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝒫1c ∈ n ∧ ¬ {x} ∈ 𝒫1c) → (𝒫1c ∪ {{x}}) ∈ (n +c 1c))
70	64, 67, 69	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → (𝒫1c ∪ {{x}}) ∈ (n +c 1c))
71		simplrr 737	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) → 𝒫1d ∈ n)
72	71	adantr 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → 𝒫1d ∈ n)
73		simprrr 741	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ¬ y ∈ d)
74		snelpw1 4147	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ({y} ∈ 𝒫1d ↔ y ∈ d)
75	73, 74	sylnibr 296	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ¬ {y} ∈ 𝒫1d)
76		snex 4112	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ {y} ∈ V
77	76	elsuci 4415	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝒫1d ∈ n ∧ ¬ {y} ∈ 𝒫1d) → (𝒫1d ∪ {{y}}) ∈ (n +c 1c))
78	72, 75, 77	syl2anc 642	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → (𝒫1d ∪ {{y}}) ∈ (n +c 1c))
79		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (m = (n +c 1c) → ((𝒫1c ∪ {{x}}) ∈ m ↔ (𝒫1c ∪ {{x}}) ∈ (n +c 1c)))
80		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (m = (n +c 1c) → ((𝒫1d ∪ {{y}}) ∈ m ↔ (𝒫1d ∪ {{y}}) ∈ (n +c 1c)))
81	79, 80	anbi12d 691	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (m = (n +c 1c) → (((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m) ↔ ((𝒫1c ∪ {{x}}) ∈ (n +c 1c) ∧ (𝒫1d ∪ {{y}}) ∈ (n +c 1c))))
82	81	rspcev 2956	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((n +c 1c) ∈ Nn ∧ ((𝒫1c ∪ {{x}}) ∈ (n +c 1c) ∧ (𝒫1d ∪ {{y}}) ∈ (n +c 1c))) → ∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m))
83	62, 70, 78, 82	syl12anc 1180	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m))
84	83	ex 423	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ k ∈ Nn ) → (((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)) → ∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m)))
85	84	ex 423	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((n ∈ Nn ∧ (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) → (k ∈ Nn → (((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)) → ∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m))))
86	85	rexlimiva 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) → (k ∈ Nn → (((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)) → ∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m))))
87		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = n → ((𝒫1c ∪ {{x}}) ∈ m ↔ (𝒫1c ∪ {{x}}) ∈ n))
88		eleq2 2414	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (m = n → ((𝒫1d ∪ {{y}}) ∈ m ↔ (𝒫1d ∪ {{y}}) ∈ n))
89	87, 88	anbi12d 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (m = n → (((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m) ↔ ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n)))
90	89	cbvrexv 2837	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∃m ∈ Nn ((𝒫1c ∪ {{x}}) ∈ m ∧ (𝒫1d ∪ {{y}}) ∈ m) ↔ ∃n ∈ Nn ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n))
91	86, 90	syl8ib 222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) → (k ∈ Nn → (((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)) → ∃n ∈ Nn ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n))))
92	91	com12 27	. . . . . . . . . . . . . . . . . . 19 ⊢ (k ∈ Nn → (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) → (((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d)) → ∃n ∈ Nn ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n))))
93	92	imp31 421	. . . . . . . . . . . . . . . . . 18 ⊢ (((k ∈ Nn ∧ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ∃n ∈ Nn ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n))
94		pw1eq 4144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (a = (c ∪ {x}) → 𝒫1a = 𝒫1(c ∪ {x}))
95		pw1un 4164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝒫1(c ∪ {x}) = (𝒫1c ∪ 𝒫1{x})
96	55	pw1sn 4166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝒫1{x} = {{x}}
97	96	uneq2i 3416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝒫1c ∪ 𝒫1{x}) = (𝒫1c ∪ {{x}})
98	95, 97	eqtri 2373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝒫1(c ∪ {x}) = (𝒫1c ∪ {{x}})
99	94, 98	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (a = (c ∪ {x}) → 𝒫1a = (𝒫1c ∪ {{x}}))
100	99	eleq1d 2419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (a = (c ∪ {x}) → (𝒫1a ∈ n ↔ (𝒫1c ∪ {{x}}) ∈ n))
101		pw1eq 4144	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (b = (d ∪ {y}) → 𝒫1b = 𝒫1(d ∪ {y}))
102		pw1un 4164	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝒫1(d ∪ {y}) = (𝒫1d ∪ 𝒫1{y})
103	57	pw1sn 4166	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝒫1{y} = {{y}}
104	103	uneq2i 3416	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝒫1d ∪ 𝒫1{y}) = (𝒫1d ∪ {{y}})
105	102, 104	eqtri 2373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝒫1(d ∪ {y}) = (𝒫1d ∪ {{y}})
106	101, 105	syl6eq 2401	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (b = (d ∪ {y}) → 𝒫1b = (𝒫1d ∪ {{y}}))
107	106	eleq1d 2419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (b = (d ∪ {y}) → (𝒫1b ∈ n ↔ (𝒫1d ∪ {{y}}) ∈ n))
108	100, 107	bi2anan9 843	. . . . . . . . . . . . . . . . . . 19 ⊢ ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ((𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n)))
109	108	rexbidv 2636	. . . . . . . . . . . . . . . . . 18 ⊢ ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → (∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn ((𝒫1c ∪ {{x}}) ∈ n ∧ (𝒫1d ∪ {{y}}) ∈ n)))
110	93, 109	syl5ibrcom 213	. . . . . . . . . . . . . . . . 17 ⊢ (((k ∈ Nn ∧ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ ((c ∈ k ∧ d ∈ k) ∧ (¬ x ∈ c ∧ ¬ y ∈ d))) → ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
111	60, 110	sylan2b 461	. . . . . . . . . . . . . . . 16 ⊢ (((k ∈ Nn ∧ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ ((c ∈ k ∧ d ∈ k) ∧ (x ∈ ∼ c ∧ y ∈ ∼ d))) → ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
112	111	expr 598	. . . . . . . . . . . . . . 15 ⊢ (((k ∈ Nn ∧ ∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n)) ∧ (c ∈ k ∧ d ∈ k)) → ((x ∈ ∼ c ∧ y ∈ ∼ d) → ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))))
113	112	anasss 628	. . . . . . . . . . . . . 14 ⊢ ((k ∈ Nn ∧ (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) ∧ (c ∈ k ∧ d ∈ k))) → ((x ∈ ∼ c ∧ y ∈ ∼ d) → ((a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))))
114	113	rexlimdvv 2745	. . . . . . . . . . . . 13 ⊢ ((k ∈ Nn ∧ (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) ∧ (c ∈ k ∧ d ∈ k))) → (∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
115	114	exp32 588	. . . . . . . . . . . 12 ⊢ (k ∈ Nn → (∃n ∈ Nn (𝒫1c ∈ n ∧ 𝒫1d ∈ n) → ((c ∈ k ∧ d ∈ k) → (∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))))
116	54, 115	sylan9r 639	. . . . . . . . . . 11 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → ((c ∈ k ∧ d ∈ k) → ((c ∈ k ∧ d ∈ k) → (∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))))
117	116	pm2.43d 44	. . . . . . . . . 10 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → ((c ∈ k ∧ d ∈ k) → (∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))))
118	117	rexlimdvv 2745	. . . . . . . . 9 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → (∃c ∈ k ∃d ∈ k ∃x ∈ ∼ c∃y ∈ ∼ d(a = (c ∪ {x}) ∧ b = (d ∪ {y})) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
119	44, 118	syl5bi 208	. . . . . . . 8 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → ((a ∈ (k +c 1c) ∧ b ∈ (k +c 1c)) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
120	119	exp3a 425	. . . . . . 7 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → (a ∈ (k +c 1c) → (b ∈ (k +c 1c) → ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))))
121	36, 37, 120	ralrimd 2703	. . . . . 6 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → (a ∈ (k +c 1c) → ∀b ∈ (k +c 1c)∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
122	33, 121	ralrimi 2696	. . . . 5 ⊢ ((k ∈ Nn ∧ ∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)) → ∀a ∈ (k +c 1c)∀b ∈ (k +c 1c)∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
123	122	ex 423	. . . 4 ⊢ (k ∈ Nn → (∀a ∈ k ∀b ∈ k ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) → ∀a ∈ (k +c 1c)∀b ∈ (k +c 1c)∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n)))
124	1, 3, 5, 7, 9, 30, 123	finds 4412	. . 3 ⊢ (M ∈ Nn → ∀a ∈ M ∀b ∈ M ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n))
125		pw1eq 4144	. . . . . . 7 ⊢ (a = A → 𝒫1a = 𝒫1A)
126	125	eleq1d 2419	. . . . . 6 ⊢ (a = A → (𝒫1a ∈ n ↔ 𝒫1A ∈ n))
127	126	anbi1d 685	. . . . 5 ⊢ (a = A → ((𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ (𝒫1A ∈ n ∧ 𝒫1b ∈ n)))
128	127	rexbidv 2636	. . . 4 ⊢ (a = A → (∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1b ∈ n)))
129		pw1eq 4144	. . . . . . 7 ⊢ (b = B → 𝒫1b = 𝒫1B)
130	129	eleq1d 2419	. . . . . 6 ⊢ (b = B → (𝒫1b ∈ n ↔ 𝒫1B ∈ n))
131	130	anbi2d 684	. . . . 5 ⊢ (b = B → ((𝒫1A ∈ n ∧ 𝒫1b ∈ n) ↔ (𝒫1A ∈ n ∧ 𝒫1B ∈ n)))
132	131	rexbidv 2636	. . . 4 ⊢ (b = B → (∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1b ∈ n) ↔ ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1B ∈ n)))
133	128, 132	rspc2v 2962	. . 3 ⊢ ((A ∈ M ∧ B ∈ M) → (∀a ∈ M ∀b ∈ M ∃n ∈ Nn (𝒫1a ∈ n ∧ 𝒫1b ∈ n) → ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1B ∈ n)))
134	124, 133	syl5com 26	. 2 ⊢ (M ∈ Nn → ((A ∈ M ∧ B ∈ M) → ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1B ∈ n)))
135	134	3impib 1149	1 ⊢ ((M ∈ Nn ∧ A ∈ M ∧ B ∈ M) → ∃n ∈ Nn (𝒫1A ∈ n ∧ 𝒫1B ∈ n))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  ∀wral 2615  ∃wrex 2616   ∼ ccompl 3206   ∪ cun 3208  ∅c0 3551  {csn 3738  1_cc1c 4135  𝒫₁cpw1 4136   Nn cnnc 4374  0_cc0c 4375   +_c cplc 4376

This proof depends on 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-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 proof 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 used by:  nnpw1ex  4485  tfinpw1  4495  pw1fin  6170