Theorem nninfisol 7323

Description: Finite elements of ℕ_∞ are isolated. That is, given a natural number and any element of ℕ_∞, it is decidable whether the natural number (when converted to an element of ℕ_∞) is equal to the given element of ℕ_∞. Stated in an online post by Martin Escardo. One way to understand this theorem is that you do not need to look at an unbounded number of elements of the sequence 𝑋 to decide whether it is equal to 𝑁 (in fact, you only need to look at two elements and 𝑁 tells you where to look).

By contrast, the point at infinity being isolated is equivalent to the Weak Limited Principle of Omniscience (WLPO) (nninfinfwlpo 7370). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)

Assertion

Ref	Expression
nninfisol	⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Distinct variable groups:   𝑖,𝑁   𝑖,𝑋

Proof of Theorem nninfisol

Step	Hyp	Ref	Expression
1		simpllr 534	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → 𝑋 ∈ ℕ∞)
2		simplr 528	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → (𝑋‘𝑁) = ∅)
3		simplll 533	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → 𝑁 ∈ ω)
4		simpr 110	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → 𝑁 = ∅)
5	1, 2, 3, 4	nninfisollem0 7320	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
6		simp-4r 542	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → 𝑋 ∈ ℕ∞)
7		simpllr 534	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → (𝑋‘𝑁) = ∅)
8		simp-4l 541	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → 𝑁 ∈ ω)
9		simpr 110	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → ¬ 𝑁 = ∅)
10	9	neqned 2407	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → 𝑁 ≠ ∅)
11	10	adantr 276	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → 𝑁 ≠ ∅)
12		simpr 110	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → (𝑋‘∪ 𝑁) = ∅)
13	6, 7, 8, 11, 12	nninfisollemne 7321	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
14		simp-4r 542	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → 𝑋 ∈ ℕ∞)
15		simpllr 534	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → (𝑋‘𝑁) = ∅)
16		simp-4l 541	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → 𝑁 ∈ ω)
17	10	adantr 276	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → 𝑁 ≠ ∅)
18		simpr 110	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → (𝑋‘∪ 𝑁) = 1o)
19	14, 15, 16, 17, 18	nninfisollemeq 7322	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
20		nninff 7312	. . . . . . . . 9 ⊢ (𝑋 ∈ ℕ∞ → 𝑋:ω⟶2o)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → 𝑋:ω⟶2o)
22		nnpredcl 4719	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ∪ 𝑁 ∈ ω)
24	21, 23	ffvelcdmd 5779	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ 2o)
25		df2o3 6592	. . . . . . 7 ⊢ 2o = {∅, 1o}
26	24, 25	eleqtrdi 2322	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ {∅, 1o})
27		elpri 3690	. . . . . 6 ⊢ ((𝑋‘∪ 𝑁) ∈ {∅, 1o} → ((𝑋‘∪ 𝑁) = ∅ ∨ (𝑋‘∪ 𝑁) = 1o))
28	26, 27	syl 14	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ((𝑋‘∪ 𝑁) = ∅ ∨ (𝑋‘∪ 𝑁) = 1o))
29	28	ad2antrr 488	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → ((𝑋‘∪ 𝑁) = ∅ ∨ (𝑋‘∪ 𝑁) = 1o))
30	13, 19, 29	mpjaodan 803	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
31		nndceq0 4714	. . . . 5 ⊢ (𝑁 ∈ ω → DECID 𝑁 = ∅)
32		exmiddc 841	. . . . 5 ⊢ (DECID 𝑁 = ∅ → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
33	31, 32	syl 14	. . . 4 ⊢ (𝑁 ∈ ω → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
34	33	ad2antrr 488	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
35	5, 30, 34	mpjaodan 803	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
36		1n0 6595	. . . . . 6 ⊢ 1o ≠ ∅
37	36	neii 2402	. . . . 5 ⊢ ¬ 1o = ∅
38		simpr 110	. . . . . . . 8 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
39	38	fveq1d 5637	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = (𝑋‘𝑁))
40		eqid 2229	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
41		eleq1 2292	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ 𝑁 ↔ 𝑁 ∈ 𝑁))
42	41	ifbid 3625	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑁 ∈ 𝑁, 1o, ∅))
43		id 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → 𝑁 ∈ ω)
44		nnord 4708	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → Ord 𝑁)
45		ordirr 4638	. . . . . . . . . . . . 13 ⊢ (Ord 𝑁 → ¬ 𝑁 ∈ 𝑁)
46	44, 45	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → ¬ 𝑁 ∈ 𝑁)
47	46	iffalsed 3613	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) = ∅)
48		peano1 4690	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
49	47, 48	eqeltrdi 2320	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) ∈ ω)
50	40, 42, 43, 49	fvmptd3 5736	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = if(𝑁 ∈ 𝑁, 1o, ∅))
51	50, 47	eqtrd 2262	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = ∅)
52	51	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = ∅)
53		simplr 528	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘𝑁) = 1o)
54	39, 52, 53	3eqtr3rd 2271	. . . . . 6 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → 1o = ∅)
55	54	ex 115	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 → 1o = ∅))
56	37, 55	mtoi 668	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
57	56	olcd 739	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
58		df-dc 840	. . 3 ⊢ (DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ↔ ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋 ∨ ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋))
59	57, 58	sylibr 134	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
60		simpl 109	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → 𝑁 ∈ ω)
61	21, 60	ffvelcdmd 5779	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ 2o)
62	61, 25	eleqtrdi 2322	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ {∅, 1o})
63		elpri 3690	. . 3 ⊢ ((𝑋‘𝑁) ∈ {∅, 1o} → ((𝑋‘𝑁) = ∅ ∨ (𝑋‘𝑁) = 1o))
64	62, 63	syl 14	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ((𝑋‘𝑁) = ∅ ∨ (𝑋‘𝑁) = 1o))
65	35, 59, 64	mpjaodan 803	1 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 713  DECID wdc 839   = wceq 1395   ∈ wcel 2200   ≠ wne 2400  ∅c0 3492  ifcif 3603  {cpr 3668  ∪ cuni 3891   ↦ cmpt 4148  Ord word 4457  ωcom 4686  ⟶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: (None)