Theorem nninfisol 7463

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 7510). (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 540	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → 𝑋 ∈ ℕ∞)
2		simplr 533	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → (𝑋‘𝑁) = ∅)
3		simplll 539	. . . 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 7460	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
6		simp-4r 548	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → 𝑋 ∈ ℕ∞)
7		simpllr 540	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → (𝑋‘𝑁) = ∅)
8		simp-4l 547	. . . . 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 2427	. . . . . 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 7461	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
14		simp-4r 548	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → 𝑋 ∈ ℕ∞)
15		simpllr 540	. . . . 5 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → (𝑋‘𝑁) = ∅)
16		simp-4l 547	. . . . 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 7462	. . . 4 ⊢ (((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) ∧ (𝑋‘∪ 𝑁) = 1o) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
20		nninff 7452	. . . . . . . . 9 ⊢ (𝑋 ∈ ℕ∞ → 𝑋:ω⟶2o)
21	20	adantl 277	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → 𝑋:ω⟶2o)
22		nnpredcl 4765	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ∪ 𝑁 ∈ ω)
23	22	adantr 276	. . . . . . . 8 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → ∪ 𝑁 ∈ ω)
24	21, 23	ffvelcdmd 5835	. . . . . . 7 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ 2o)
25		df2o3 6692	. . . . . . 7 ⊢ 2o = {∅, 1o}
26	24, 25	eleqtrdi 2331	. . . . . 6 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘∪ 𝑁) ∈ {∅, 1o})
27		elpri 3728	. . . . . 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 492	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → ((𝑋‘∪ 𝑁) = ∅ ∨ (𝑋‘∪ 𝑁) = 1o))
30	13, 19, 29	mpjaodan 810	. . 3 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) ∧ ¬ 𝑁 = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
31		nndceq0 4760	. . . . 5 ⊢ (𝑁 ∈ ω → DECID 𝑁 = ∅)
32		exmiddc 848	. . . . 5 ⊢ (DECID 𝑁 = ∅ → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
33	31, 32	syl 14	. . . 4 ⊢ (𝑁 ∈ ω → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
34	33	ad2antrr 492	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) → (𝑁 = ∅ ∨ ¬ 𝑁 = ∅))
35	5, 30, 34	mpjaodan 810	. 2 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = ∅) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
36		1n0 6695	. . . . . 6 ⊢ 1o ≠ ∅
37	36	neii 2422	. . . . 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 5692	. . . . . . 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 2238	. . . . . . . . . 10 ⊢ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))
41		eleq1 2301	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑁 → (𝑖 ∈ 𝑁 ↔ 𝑁 ∈ 𝑁))
42	41	ifbid 3659	. . . . . . . . . 10 ⊢ (𝑖 = 𝑁 → if(𝑖 ∈ 𝑁, 1o, ∅) = if(𝑁 ∈ 𝑁, 1o, ∅))
43		id 19	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → 𝑁 ∈ ω)
44		nnord 4754	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ω → Ord 𝑁)
45		ordirr 4684	. . . . . . . . . . . . 13 ⊢ (Ord 𝑁 → ¬ 𝑁 ∈ 𝑁)
46	44, 45	syl 14	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ω → ¬ 𝑁 ∈ 𝑁)
47	46	iffalsed 3647	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) = ∅)
48		peano1 4736	. . . . . . . . . . 11 ⊢ ∅ ∈ ω
49	47, 48	eqeltrdi 2329	. . . . . . . . . 10 ⊢ (𝑁 ∈ ω → if(𝑁 ∈ 𝑁, 1o, ∅) ∈ ω)
50	40, 42, 43, 49	fvmptd3 5793	. . . . . . . . 9 ⊢ (𝑁 ∈ ω → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = if(𝑁 ∈ 𝑁, 1o, ∅))
51	50, 47	eqtrd 2271	. . . . . . . 8 ⊢ (𝑁 ∈ ω → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = ∅)
52	51	ad3antrrr 496	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → ((𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅))‘𝑁) = ∅)
53		simplr 533	. . . . . . 7 ⊢ ((((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) ∧ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋) → (𝑋‘𝑁) = 1o)
54	39, 52, 53	3eqtr3rd 2280	. . . . . 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 674	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) ∧ (𝑋‘𝑁) = 1o) → ¬ (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)
57	56	olcd 746	. . 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 847	. . 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 5835	. . . 4 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ 2o)
62	61, 25	eleqtrdi 2331	. . 3 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → (𝑋‘𝑁) ∈ {∅, 1o})
63		elpri 3728	. . 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 810	1 ⊢ ((𝑁 ∈ ω ∧ 𝑋 ∈ ℕ∞) → DECID (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑁, 1o, ∅)) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∅c0 3520  ifcif 3635  {cpr 3706  ∪ cuni 3930   ↦ cmpt 4187  Ord word 4502  ωcom 4732  ⟶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: (None)