nnnninfex - Mathbox for Jim Kingdon

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Theorem nnnninfex 16624

Description: If an element of ℕ_∞ has a value of zero somewhere, then it is the mapping of a natural number. (Contributed by Jim Kingdon, 4-Aug-2022.)

**Hypotheses**
Ref	Expression
nnnninfex.p	⊢ (𝜑 → 𝑃 ∈ ℕ_∞)
nnnninfex.n	⊢ (𝜑 → 𝑁 ∈ ω)
nnnninfex.0	⊢ (𝜑 → (𝑃‘𝑁) = ∅)

**Assertion**
Ref	Expression
nnnninfex	⊢ (𝜑 → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))

Distinct variable group: 𝑃,𝑖,𝑛 Allowed substitution hints: 𝜑(𝑖,𝑛) 𝑁(𝑖,𝑛)

Proof of Theorem nnnninfex Dummy variables 𝑤 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnnninfex.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ω)
2		nnnninfex.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℕ_∞)
3		nnnninfex.0	. . 3 ⊢ (𝜑 → (𝑃‘𝑁) = ∅)
4	2, 3	jca 306	. 2 ⊢ (𝜑 → (𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑁) = ∅))
5		fveqeq2 5648	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘∅) = ∅))
6	5	anbi2d 464	. . . 4 ⊢ (𝑤 = ∅ → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) ↔ (𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅)))
7	6	imbi1d 231	. . 3 ⊢ (𝑤 = ∅ → (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
8		fveqeq2 5648	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑘) = ∅))
9	8	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑘 → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) ↔ (𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑘) = ∅)))
10	9	imbi1d 231	. . 3 ⊢ (𝑤 = 𝑘 → (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
11		fveqeq2 5648	. . . . 5 ⊢ (𝑤 = suc 𝑘 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘suc 𝑘) = ∅))
12	11	anbi2d 464	. . . 4 ⊢ (𝑤 = suc 𝑘 → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) ↔ (𝑃 ∈ ℕ_∞ ∧ (𝑃‘suc 𝑘) = ∅)))
13	12	imbi1d 231	. . 3 ⊢ (𝑤 = suc 𝑘 → (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
14		fveqeq2 5648	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑁) = ∅))
15	14	anbi2d 464	. . . 4 ⊢ (𝑤 = 𝑁 → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) ↔ (𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑁) = ∅)))
16	15	imbi1d 231	. . 3 ⊢ (𝑤 = 𝑁 → (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑤) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑁) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
17		peano1 4692	. . . 4 ⊢ ∅ ∈ ω
18		simpll 527	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → 𝑃 ∈ ℕ_∞)
19	17	a1i 9	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ∅ ∈ ω)
20		simpr 110	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → 𝑗 ∈ ω)
21		0ss 3533	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑗
22	21	a1i 9	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ∅ ⊆ 𝑗)
23		simplr 529	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘∅) = ∅)
24	18, 19, 20, 22, 23	nninfninc 7321	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ∅)
25		noel 3498	. . . . . . . . . 10 ⊢ ¬ 𝑖 ∈ ∅
26	25	iffalsei 3614	. . . . . . . . 9 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) = ∅
27	26	mpteq2i 4176	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ ∅)
28		eqidd 2232	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ∅ = ∅)
29	27, 28, 20, 19	fvmptd3 5740	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗) = ∅)
30	24, 29	eqtr4d 2267	. . . . . 6 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
31	30	ralrimiva 2605	. . . . 5 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
32		nninff 7320	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃:ω⟶2_o)
33	32	ffnd 5483	. . . . . . 7 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃 Fn ω)
34	33	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → 𝑃 Fn ω)
35		1oex 6589	. . . . . . . 8 ⊢ 1_o ∈ V
36		0ex 4216	. . . . . . . 8 ⊢ ∅ ∈ V
37	35, 36	ifex 4583	. . . . . . 7 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) ∈ V
38		eqid 2231	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))
39	37, 38	fnmpti 5461	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) Fn ω
40		eqfnfv 5744	. . . . . 6 ⊢ ((𝑃 Fn ω ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) Fn ω) → (𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) ↔ ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗)))
41	34, 39, 40	sylancl 413	. . . . 5 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → (𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) ↔ ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗)))
42	31, 41	mpbird 167	. . . 4 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)))
43		eleq2 2295	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅))
44	43	ifbid 3627	. . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ ∅, 1_o, ∅))
45	44	mpteq2dv 4180	. . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)))
46	45	rspceeqv 2928	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
47	17, 42, 46	sylancr 414	. . 3 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
48		simpr 110	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → (𝑃‘𝑘) = ∅)
49		simpllr 536	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))))
50	48, 49	mpd 13	. . . . . . 7 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
51		simpl 109	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → 𝑘 ∈ ω)
52	51	ad3antrrr 492	. . . . . . . . 9 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑘 ∈ ω)
53		peano2 4693	. . . . . . . . 9 ⊢ (𝑘 ∈ ω → suc 𝑘 ∈ ω)
54	52, 53	syl 14	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → suc 𝑘 ∈ ω)
55		simpllr 536	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑃 ∈ ℕ_∞)
56	53	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → suc 𝑘 ∈ ω)
57		nnord 4710	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ω → Ord 𝑘)
58		ordtr 4475	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑘 → Tr 𝑘)
59	57, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → Tr 𝑘)
60		unisucg 4511	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
61	59, 60	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
62	61	fveq2d 5643	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
63	62	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
64		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘𝑘) = 1_o)
65	63, 64	eqtrd 2264	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘∪ suc 𝑘) = 1_o)
66		simplr 529	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘suc 𝑘) = ∅)
67	55, 56, 65, 66	nnnninfeq2 7327	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
68	67	adantllr 481	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
69		eleq2 2295	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ suc 𝑘))
70	69	ifbid 3627	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑘 → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ suc 𝑘, 1_o, ∅))
71	70	mpteq2dv 4180	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
72	71	rspceeqv 2928	. . . . . . . 8 ⊢ ((suc 𝑘 ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
73	54, 68, 72	syl2anc 411	. . . . . . 7 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
74	32	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → 𝑃:ω⟶2_o)
75	74, 51	ffvelcdmd 5783	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ 2_o)
76		df2o3 6596	. . . . . . . . . 10 ⊢ 2_o = {∅, 1_o}
77	75, 76	eleqtrdi 2324	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ {∅, 1_o})
78		elpri 3692	. . . . . . . . 9 ⊢ ((𝑃‘𝑘) ∈ {∅, 1_o} → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1_o))
79	77, 78	syl 14	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1_o))
80	79	ad2antrr 488	. . . . . . 7 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1_o))
81	50, 73, 80	mpjaodan 805	. . . . . 6 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
82	81	exp41 370	. . . . 5 ⊢ (𝑘 ∈ ω → (𝑃 ∈ ℕ_∞ → (((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))))))
83	82	a2d 26	. . . 4 ⊢ (𝑘 ∈ ω → ((𝑃 ∈ ℕ_∞ → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))) → (𝑃 ∈ ℕ_∞ → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))))))
84		impexp 263	. . . 4 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ (𝑃 ∈ ℕ_∞ → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
85		impexp 263	. . . 4 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) ↔ (𝑃 ∈ ℕ_∞ → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
86	83, 84, 85	3imtr4g 205	. . 3 ⊢ (𝑘 ∈ ω → (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))) → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))))
87	7, 10, 13, 16, 47, 86	finds 4698	. 2 ⊢ (𝑁 ∈ ω → ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘𝑁) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅))))
88	1, 4, 87	sylc 62	1 ⊢ (𝜑 → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 = wceq 1397 ∈ wcel 2202 ∀wral 2510 ∃wrex 2511 ⊆ wss 3200 ∅c0 3494 ifcif 3605 {cpr 3670 ∪ cuni 3893 ↦ cmpt 4150 Tr wtr 4187 Ord word 4459 suc csuc 4462 ωcom 4688 Fn wfn 5321 ⟶wf 5322 ‘cfv 5326 1_oc1o 6574 2_oc2o 6575 ℕ_∞xnninf 7317

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1o 6581 df-2o 6582 df-map 6818 df-nninf 7318

This theorem is referenced by: nninfnfiinf 16625

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