Theorem nninfisol 7296

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 7343). (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 7293	. . 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 7294	. . . 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 7295	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
20		nninff 7285	. . . . . . . . 9 ⊢ (𝑋 ∈ ℕ∞ → 𝑋:ω⟶2o)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → 𝑋:ω⟶2o)
22		nnpredcl 4714	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ∪ 𝑁 ∈ ω)
24	21, 23	ffvelcdmd 5770	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ 2o)
25		df2o3 6574	. . . . . . 7 ⊢ 2o = {∅, 1o}
26	24, 25	eleqtrdi 2322	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ {∅, 1o})
27		elpri 3689	. . . . . 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 4709	. . . . 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 6576	. . . . . 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 5628	. . . . . . 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 3624	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑁 ∈ 𝑁, 1o, ∅))
43		id 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → 𝑁 ∈ ω)
44		nnord 4703	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → Ord 𝑁)
45		ordirr 4633	. . . . . . . . . . . . 13 ⊢ (Ord 𝑁 → ¬ 𝑁 ∈ 𝑁)
46	44, 45	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → ¬ 𝑁 ∈ 𝑁)
47	46	iffalsed 3612	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) = ∅)
48		peano1 4685	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
49	47, 48	eqeltrdi 2320	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) ∈ ω)
50	40, 42, 43, 49	fvmptd3 5727	. . . . . . . . 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 5770	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ 2o)
62	61, 25	eleqtrdi 2322	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ {∅, 1o})
63		elpri 3689	. . 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 3491  ifcif 3602  {cpr 3667  ∪ cuni 3887   ↦ cmpt 4144  Ord word 4452  ωcom 4681  ⟶wf 5313  ‘cfv 5317  1_oc1o 6553  2_oc2o 6554  ℕ_∞xnninf 7282

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 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1o 6560  df-2o 6561  df-map 6795  df-nninf 7283

This theorem is referenced by: (None)