nnnninfex - Mathbox for Jim Kingdon

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Theorem nnnninfex 16926

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 5684	. . . . 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 5684	. . . . 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 5684	. . . . 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 5684	. . . . 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 4721	. . . 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 3551	. . . . . . . . 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 7427	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ∅)
25		noel 3516	. . . . . . . . . 10 ⊢ ¬ 𝑖 ∈ ∅
26	25	iffalsei 3635	. . . . . . . . 9 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) = ∅
27	26	mpteq2i 4202	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ ∅)
28		eqidd 2235	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ∅ = ∅)
29	27, 28, 20, 19	fvmptd3 5776	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗) = ∅)
30	24, 29	eqtr4d 2270	. . . . . 6 ⊢ (((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
31	30	ralrimiva 2617	. . . . 5 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))‘𝑗))
32		nninff 7426	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃:ω⟶2_o)
33	32	ffnd 5514	. . . . . . 7 ⊢ (𝑃 ∈ ℕ_∞ → 𝑃 Fn ω)
34	33	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℕ_∞ ∧ (𝑃‘∅) = ∅) → 𝑃 Fn ω)
35		1oex 6668	. . . . . . . 8 ⊢ 1_o ∈ V
36		0ex 4242	. . . . . . . 8 ⊢ ∅ ∈ V
37	35, 36	ifex 4612	. . . . . . 7 ⊢ if(𝑖 ∈ ∅, 1_o, ∅) ∈ V
38		eqid 2234	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅))
39	37, 38	fnmpti 5492	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)) Fn ω
40		eqfnfv 5780	. . . . . 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 2298	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅))
44	43	ifbid 3648	. . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ ∅, 1_o, ∅))
45	44	mpteq2dv 4206	. . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1_o, ∅)))
46	45	rspceeqv 2942	. . . 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 4722	. . . . . . . . 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 4739	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ω → Ord 𝑘)
58		ordtr 4504	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑘 → Tr 𝑘)
59	57, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → Tr 𝑘)
60		unisucg 4540	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
61	59, 60	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
62	61	fveq2d 5679	. . . . . . . . . . . 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 2267	. . . . . . . . . 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 7433	. . . . . . . . 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 2298	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ suc 𝑘))
70	69	ifbid 3648	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑘 → if(𝑖 ∈ 𝑛, 1_o, ∅) = if(𝑖 ∈ suc 𝑘, 1_o, ∅))
71	70	mpteq2dv 4206	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1_o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1_o, ∅)))
72	71	rspceeqv 2942	. . . . . . . 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 5818	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ 2_o)
76		df2o3 6675	. . . . . . . . . 10 ⊢ 2_o = {∅, 1_o}
77	75, 76	eleqtrdi 2327	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ_∞) → (𝑃‘𝑘) ∈ {∅, 1_o})
78		elpri 3717	. . . . . . . . 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 806	. . . . . 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 4727	. 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 716 = wceq 1398 ∈ wcel 2205 ∀wral 2522 ∃wrex 2523 ⊆ wss 3214 ∅c0 3512 ifcif 3624 {cpr 3695 ∪ cuni 3919 ↦ cmpt 4176 Tr wtr 4213 Ord word 4488 suc csuc 4491 ωcom 4717 Fn wfn 5352 ⟶wf 5353 ‘cfv 5357 1_oc1o 6653 2_oc2o 6654 ℕ_∞xnninf 7423

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-nul 4241 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-iinf 4715

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-csb 3142 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-nul 3513 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-int 3955 df-br 4115 df-opab 4177 df-mpt 4178 df-tr 4214 df-id 4419 df-iord 4492 df-on 4494 df-suc 4497 df-iom 4718 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-rn 4765 df-iota 5317 df-fun 5359 df-fn 5360 df-f 5361 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-1o 6660 df-2o 6661 df-map 6897 df-nninf 7424

This theorem is referenced by: nninfnfiinf 16927

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