Theorem nnnninfex 16560

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, 1o, ∅)))

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 5644	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
8		fveqeq2 5644	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
11		fveqeq2 5644	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
14		fveqeq2 5644	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑁) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
17		peano1 4690	. . . 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 3531	. . . . . . . . 9 ⊢ ∅ ⊆ 𝑗
22	21	a1i 9	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ∅ ⊆ 𝑗)
23		simplr 528	. . . . . . . 8 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘∅) = ∅)
24	18, 19, 20, 22, 23	nninfninc 7313	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ∅)
25		noel 3496	. . . . . . . . . 10 ⊢ ¬ 𝑖 ∈ ∅
26	25	iffalsei 3612	. . . . . . . . 9 ⊢ if(𝑖 ∈ ∅, 1o, ∅) = ∅
27	26	mpteq2i 4174	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ ∅)
28		eqidd 2230	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ∅ = ∅)
29	27, 28, 20, 19	fvmptd3 5736	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗) = ∅)
30	24, 29	eqtr4d 2265	. . . . . 6 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗))
31	30	ralrimiva 2603	. . . . 5 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗))
32		nninff 7312	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ∞ → 𝑃:ω⟶2o)
33	32	ffnd 5480	. . . . . . 7 ⊢ (𝑃 ∈ ℕ∞ → 𝑃 Fn ω)
34	33	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → 𝑃 Fn ω)
35		1oex 6585	. . . . . . . 8 ⊢ 1o ∈ V
36		0ex 4214	. . . . . . . 8 ⊢ ∅ ∈ V
37	35, 36	ifex 4581	. . . . . . 7 ⊢ if(𝑖 ∈ ∅, 1o, ∅) ∈ V
38		eqid 2229	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))
39	37, 38	fnmpti 5458	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) Fn ω
40		eqfnfv 5740	. . . . . 6 ⊢ ((𝑃 Fn ω ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) Fn ω) → (𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) ↔ ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗)))
41	34, 39, 40	sylancl 413	. . . . 5 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → (𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) ↔ ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗)))
42	31, 41	mpbird 167	. . . 4 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)))
43		eleq2 2293	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅))
44	43	ifbid 3625	. . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ ∅, 1o, ∅))
45	44	mpteq2dv 4178	. . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)))
46	45	rspceeqv 2926	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
47	17, 42, 46	sylancr 414	. . 3 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
48		simpr 110	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → (𝑃‘𝑘) = ∅)
49		simpllr 534	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
50	48, 49	mpd 13	. . . . . . 7 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
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, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑘 ∈ ω)
53		peano2 4691	. . . . . . . . 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, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → suc 𝑘 ∈ ω)
55		simpllr 534	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑃 ∈ ℕ∞)
56	53	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → suc 𝑘 ∈ ω)
57		nnord 4708	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ω → Ord 𝑘)
58		ordtr 4473	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑘 → Tr 𝑘)
59	57, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → Tr 𝑘)
60		unisucg 4509	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
61	59, 60	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
62	61	fveq2d 5639	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
63	62	ad3antrrr 492	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
64		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘𝑘) = 1o)
65	63, 64	eqtrd 2262	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘∪ suc 𝑘) = 1o)
66		simplr 528	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘suc 𝑘) = ∅)
67	55, 56, 65, 66	nnnninfeq2 7319	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅)))
68	67	adantllr 481	. . . . . . . 8 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅)))
69		eleq2 2293	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ suc 𝑘))
70	69	ifbid 3625	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑘 → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ suc 𝑘, 1o, ∅))
71	70	mpteq2dv 4178	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅)))
72	71	rspceeqv 2926	. . . . . . . 8 ⊢ ((suc 𝑘 ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
73	54, 68, 72	syl2anc 411	. . . . . . 7 ⊢ (((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
74	32	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → 𝑃:ω⟶2o)
75	74, 51	ffvelcdmd 5779	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → (𝑃‘𝑘) ∈ 2o)
76		df2o3 6592	. . . . . . . . . 10 ⊢ 2o = {∅, 1o}
77	75, 76	eleqtrdi 2322	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → (𝑃‘𝑘) ∈ {∅, 1o})
78		elpri 3690	. . . . . . . . 9 ⊢ ((𝑃‘𝑘) ∈ {∅, 1o} → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1o))
79	77, 78	syl 14	. . . . . . . 8 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1o))
80	79	ad2antrr 488	. . . . . . 7 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) → ((𝑃‘𝑘) = ∅ ∨ (𝑃‘𝑘) = 1o))
81	50, 73, 80	mpjaodan 803	. . . . . 6 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
82	81	exp41 370	. . . . 5 ⊢ (𝑘 ∈ ω → (𝑃 ∈ ℕ∞ → (((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))))
83	82	a2d 26	. . . 4 ⊢ (𝑘 ∈ ω → ((𝑃 ∈ ℕ∞ → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → (𝑃 ∈ ℕ∞ → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))))
84		impexp 263	. . . 4 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ↔ (𝑃 ∈ ℕ∞ → ((𝑃‘𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
85		impexp 263	. . . 4 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) ↔ (𝑃 ∈ ℕ∞ → ((𝑃‘suc 𝑘) = ∅ → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
86	83, 84, 85	3imtr4g 205	. . 3 ⊢ (𝑘 ∈ ω → (((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ((𝑃 ∈ ℕ∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
87	7, 10, 13, 16, 47, 86	finds 4696	. 2 ⊢ (𝑁 ∈ ω → ((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑁) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
88	1, 4, 87	sylc 62	1 ⊢ (𝜑 → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509   ⊆ wss 3198  ∅c0 3492  ifcif 3603  {cpr 3668  ∪ cuni 3891   ↦ cmpt 4148  Tr wtr 4185  Ord word 4457  suc csuc 4460  ωcom 4686   Fn wfn 5319  ⟶wf 5320  ‘cfv 5324  1_oc1o 6570  2_oc2o 6571  ℕ_∞xnninf 7309

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1o 6577  df-2o 6578  df-map 6814  df-nninf 7310

This theorem is referenced by:  nninfnfiinf  16561