Theorem nninfisol 7331

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 7378). (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 536	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → 𝑋 ∈ ℕ∞)
2		simplr 529	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → (𝑋‘𝑁) = ∅)
3		simplll 535	. . . 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 7328	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
6		simp-4r 544	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → 𝑋 ∈ ℕ∞)
7		simpllr 536	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → (𝑋‘𝑁) = ∅)
8		simp-4l 543	. . . . 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 2409	. . . . . 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 7329	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
14		simp-4r 544	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → 𝑋 ∈ ℕ∞)
15		simpllr 536	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → (𝑋‘𝑁) = ∅)
16		simp-4l 543	. . . . 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 7330	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
20		nninff 7320	. . . . . . . . 9 ⊢ (𝑋 ∈ ℕ∞ → 𝑋:ω⟶2o)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → 𝑋:ω⟶2o)
22		nnpredcl 4721	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ∪ 𝑁 ∈ ω)
24	21, 23	ffvelcdmd 5783	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ 2o)
25		df2o3 6596	. . . . . . 7 ⊢ 2o = {∅, 1o}
26	24, 25	eleqtrdi 2324	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ {∅, 1o})
27		elpri 3692	. . . . . 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 805	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
31		nndceq0 4716	. . . . 5 ⊢ (𝑁 ∈ ω → DECID 𝑁 = ∅)
32		exmiddc 843	. . . . 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 805	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
36		1n0 6599	. . . . . 6 ⊢ 1o ≠ ∅
37	36	neii 2404	. . . . 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 5641	. . . . . . 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 2231	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
41		eleq1 2294	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ 𝑁 ↔ 𝑁 ∈ 𝑁))
42	41	ifbid 3627	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑁 ∈ 𝑁, 1o, ∅))
43		id 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → 𝑁 ∈ ω)
44		nnord 4710	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → Ord 𝑁)
45		ordirr 4640	. . . . . . . . . . . . 13 ⊢ (Ord 𝑁 → ¬ 𝑁 ∈ 𝑁)
46	44, 45	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → ¬ 𝑁 ∈ 𝑁)
47	46	iffalsed 3615	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) = ∅)
48		peano1 4692	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
49	47, 48	eqeltrdi 2322	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) ∈ ω)
50	40, 42, 43, 49	fvmptd3 5740	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = if(𝑁 ∈ 𝑁, 1o, ∅))
51	50, 47	eqtrd 2264	. . . . . . . 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 529	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘𝑁) = 1o)
54	39, 52, 53	3eqtr3rd 2273	. . . . . 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 670	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
57	56	olcd 741	. . 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 842	. . 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 5783	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ 2o)
62	61, 25	eleqtrdi 2324	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ {∅, 1o})
63		elpri 3692	. . 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 805	1 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202   ≠ wne 2402  ∅c0 3494  ifcif 3605  {cpr 3670  ∪ cuni 3893   ↦ cmpt 4150  Ord word 4459  ωcom 4688  ⟶wf 5322  ‘cfv 5326  1_oc1o 6574  2_oc2o 6575  ℕ_∞xnninf 7317

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1o 6581  df-2o 6582  df-map 6818  df-nninf 7318

This theorem is referenced by: (None)