nnnninfex - Mathbox for Jim Kingdon

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Theorem nnnninfex 17065

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 5704	. . . . 5 ⊢ (𝑤 = ∅ → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘∅) = ∅))
6	5	anbi2d 468	. . . 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 5704	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑘) = ∅))
9	8	anbi2d 468	. . . 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 5704	. . . . 5 ⊢ (𝑤 = suc 𝑘 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘suc 𝑘) = ∅))
12	11	anbi2d 468	. . . 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 5704	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝑃‘𝑤) = ∅ ↔ (𝑃‘𝑁) = ∅))
15	14	anbi2d 468	. . . 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 4741	. . . 4 ⊢ ∅ ∈ ω
18		simpll 531	. . . . . . . 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 3561	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑗
22	21	a1i 9	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ∅ ⊆ 𝑗)
23		simplr 533	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘∅) = ∅)
24	18, 19, 20, 22, 23	nninfninc 7463	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ∅)
25		noel 3525	. . . . . . . . . 10 ⊢ ¬ 𝑖 ∈ ∅
26	25	iffalsei 3649	. . . . . . . . 9 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) = ∅
27	26	mpteq2i 4218	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ ∅)
28		eqidd 2239	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ∅ = ∅)
29	27, 28, 20, 19	fvmptd3 5799	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗) = ∅)
30	24, 29	eqtr4d 2274	. . . . . 6 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
31	30	ralrimiva 2623	. . . . 5 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
32		nninff 7462	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃:ω⟶2_o)
33	32	ffnd 5534	. . . . . . 7 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃 Fn ω)
34	33	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → 𝑃 Fn ω)
35		1oex 6695	. . . . . . . 8 ⊢ 1_o ∈ V
36		0ex 4260	. . . . . . . 8 ⊢ ∅ ∈ V
37	35, 36	ifex 4632	. . . . . . 7 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) ∈ V
38		eqid 2238	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))
39	37, 38	fnmpti 5512	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) Fn ω
40		eqfnfv 5806	. . . . . 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 417	. . . . 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 2302	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅))
44	43	ifbid 3662	. . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ ∅, 1_o, ∅))
45	44	mpteq2dv 4222	. . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)))
46	45	rspceeqv 2948	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)))
47	17, 42, 46	sylancr 418	. . 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 540	. . . . . . . 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 496	. . . . . . . . 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 4742	. . . . . . . . 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 540	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑃 ∈ ℕ_∞)
56	53	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → suc 𝑘 ∈ ω)
57		nnord 4759	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ω → Ord 𝑘)
58		ordtr 4523	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑘 → Tr 𝑘)
59	57, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → Tr 𝑘)
60		unisucg 4559	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
61	59, 60	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
62	61	fveq2d 5699	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
63	62	ad3antrrr 496	. . . . . . . . . . 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 2271	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘∪ suc 𝑘) = 1_o)
66		simplr 533	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → (𝑃‘suc 𝑘) = ∅)
67	55, 56, 65, 66	nnnninfeq2 7469	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1_o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
68	67	adantllr 485	. . . . . . . 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 2302	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ suc 𝑘))
70	69	ifbid 3662	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑘 → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ suc 𝑘, 1_o, ∅))
71	70	mpteq2dv 4222	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
72	71	rspceeqv 2948	. . . . . . . 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 415	. . . . . . 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 5844	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ 2_o)
76		df2o3 6702	. . . . . . . . . 10 ⊢ 2_o = {∅, 1_o}
77	75, 76	eleqtrdi 2331	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ {∅, 1_o})
78		elpri 3732	. . . . . . . . 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 492	. . . . . . 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 810	. . . . . 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 4747	. 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

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 720 = wceq 1402 ∈ wcel 2209 ∀wral 2528 ∃wrex 2529 ⊆ wss 3220 ∅c0 3520 ifcif 3638 {cpr 3710 ∪ cuni 3935 ↦ cmpt 4192 Tr wtr 4229 Ord word 4507 suc csuc 4510 ωcom 4737 Fn wfn 5372 ⟶wf 5373 ‘cfv 5377 1_oc1o 6680 2_oc2o 6681 ℕ_∞xnninf 7459

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-fv 5385 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1o 6687 df-2o 6688 df-map 6924 df-nninf 7460

This theorem is used by: nninfnfiinf 17066

W3C validator