Theorem nnnninfex 16970

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 5699	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
8		fveqeq2 5699	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
11		fveqeq2 5699	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘suc 𝑘) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
14		fveqeq2 5699	. . . . 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, 1o, ∅))) ↔ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘𝑁) = ∅) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
17		peano1 4736	. . . 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 7453	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ∅)
25		noel 3525	. . . . . . . . . 10 ⊢ ¬ 𝑖 ∈ ∅
26	25	iffalsei 3646	. . . . . . . . 9 ⊢ if(𝑖 ∈ ∅, 1o, ∅) = ∅
27	26	mpteq2i 4213	. . . . . . . 8 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ ∅)
28		eqidd 2239	. . . . . . . 8 ⊢ (𝑖 = 𝑗 → ∅ = ∅)
29	27, 28, 20, 19	fvmptd3 5793	. . . . . . 7 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗) = ∅)
30	24, 29	eqtr4d 2274	. . . . . 6 ⊢ (((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) ∧ 𝑗 ∈ ω) → (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗))
31	30	ralrimiva 2623	. . . . 5 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → ∀𝑗 ∈ ω (𝑃‘𝑗) = ((𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))‘𝑗))
32		nninff 7452	. . . . . . . 8 ⊢ (𝑃 ∈ ℕ∞ → 𝑃:ω⟶2o)
33	32	ffnd 5529	. . . . . . 7 ⊢ (𝑃 ∈ ℕ∞ → 𝑃 Fn ω)
34	33	adantr 276	. . . . . 6 ⊢ ((𝑃 ∈ ℕ∞ ∧ (𝑃‘∅) = ∅) → 𝑃 Fn ω)
35		1oex 6685	. . . . . . . 8 ⊢ 1o ∈ V
36		0ex 4255	. . . . . . . 8 ⊢ ∅ ∈ V
37	35, 36	ifex 4627	. . . . . . 7 ⊢ if(𝑖 ∈ ∅, 1o, ∅) ∈ V
38		eqid 2238	. . . . . . 7 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))
39	37, 38	fnmpti 5507	. . . . . 6 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)) Fn ω
40		eqfnfv 5797	. . . . . 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 417	. . . . 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 2302	. . . . . . 7 ⊢ (𝑛 = ∅ → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ ∅))
44	43	ifbid 3659	. . . . . 6 ⊢ (𝑛 = ∅ → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ ∅, 1o, ∅))
45	44	mpteq2dv 4217	. . . . 5 ⊢ (𝑛 = ∅ → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅)))
46	45	rspceeqv 2948	. . . 4 ⊢ ((∅ ∈ ω ∧ 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ ∅, 1o, ∅))) → ∃𝑛 ∈ ω 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))
47	17, 42, 46	sylancr 418	. . 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 540	. . . . . . . 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 496	. . . . . . . . 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 4737	. . . . . . . . 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 540	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑃 ∈ ℕ∞)
56	53	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → suc 𝑘 ∈ ω)
57		nnord 4754	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ω → Ord 𝑘)
58		ordtr 4518	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝑘 → Tr 𝑘)
59	57, 58	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → Tr 𝑘)
60		unisucg 4554	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ω → (Tr 𝑘 ↔ ∪ suc 𝑘 = 𝑘))
61	59, 60	mpbid 147	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ω → ∪ suc 𝑘 = 𝑘)
62	61	fveq2d 5694	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ω → (𝑃‘∪ suc 𝑘) = (𝑃‘𝑘))
63	62	ad3antrrr 496	. . . . . . . . . . 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 2271	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘∪ suc 𝑘) = 1o)
66		simplr 533	. . . . . . . . . 10 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → (𝑃‘suc 𝑘) = ∅)
67	55, 56, 65, 66	nnnninfeq2 7459	. . . . . . . . 9 ⊢ ((((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) ∧ (𝑃‘suc 𝑘) = ∅) ∧ (𝑃‘𝑘) = 1o) → 𝑃 = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅)))
68	67	adantllr 485	. . . . . . . 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 2302	. . . . . . . . . . 11 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ 𝑛 ↔ 𝑖 ∈ suc 𝑘))
70	69	ifbid 3659	. . . . . . . . . 10 ⊢ (𝑛 = suc 𝑘 → if(𝑖 ∈ 𝑛, 1o, ∅) = if(𝑖 ∈ suc 𝑘, 1o, ∅))
71	70	mpteq2dv 4217	. . . . . . . . 9 ⊢ (𝑛 = suc 𝑘 → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ suc 𝑘, 1o, ∅)))
72	71	rspceeqv 2948	. . . . . . . 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 415	. . . . . . 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 5835	. . . . . . . . . 10 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → (𝑃‘𝑘) ∈ 2o)
76		df2o3 6692	. . . . . . . . . 10 ⊢ 2o = {∅, 1o}
77	75, 76	eleqtrdi 2331	. . . . . . . . 9 ⊢ ((𝑘 ∈ ω ∧ 𝑃 ∈ ℕ∞) → (𝑃‘𝑘) ∈ {∅, 1o})
78		elpri 3728	. . . . . . . . 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 492	. . . . . . 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 810	. . . . . 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 4742	. 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 720   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529   ⊆ wss 3220  ∅c0 3520  ifcif 3635  {cpr 3706  ∪ cuni 3930   ↦ cmpt 4187  Tr wtr 4224  Ord word 4502  suc csuc 4505  ωcom 4732   Fn wfn 5367  ⟶wf 5368  ‘cfv 5372  1_oc1o 6670  2_oc2o 6671  ℕ_∞xnninf 7449

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 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 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1o 6677  df-2o 6678  df-map 6914  df-nninf 7450

This theorem is referenced by:  nninfnfiinf  16971