Theorem nninfisol 7208

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). (Contributed by BJ and Jim Kingdon, 12-Sep-2024.)

Assertion

Ref	Expression
nninfisol	|- ( ( N e. om /\ X e. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )

Distinct variable groups:   i, N   i, X

Proof of Theorem nninfisol

Step	Hyp	Ref	Expression
1		simpllr 534	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> X e. ℕ∞ )
2		simplr 528	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> ( X ` N ) = (/) )
3		simplll 533	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> N e. om )
4		simpr 110	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> N = (/) )
5	1, 2, 3, 4	nninfisollem0 7205	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
6		simp-4r 542	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> X e. ℕ∞ )
7		simpllr 534	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> ( X ` N ) = (/) )
8		simp-4l 541	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> N e. om )
9		simpr 110	. . . . . . 7 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> -. N = (/) )
10	9	neqned 2374	. . . . . 6 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> N =/= (/) )
11	10	adantr 276	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> N =/= (/) )
12		simpr 110	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> ( X ` U. N ) = (/) )
13	6, 7, 8, 11, 12	nninfisollemne 7206	. . . 4 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
14		simp-4r 542	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> X e. ℕ∞ )
15		simpllr 534	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> ( X ` N ) = (/) )
16		simp-4l 541	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> N e. om )
17	10	adantr 276	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> N =/= (/) )
18		simpr 110	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> ( X ` U. N ) = 1o )
19	14, 15, 16, 17, 18	nninfisollemeq 7207	. . . 4 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
20		nninff 7197	. . . . . . . . 9 |- ( X e. ℕ∞ -> X : om --> 2o )
21	20	adantl 277	. . . . . . . 8 |- ( ( N e. om /\ X e. ℕ∞ ) -> X : om --> 2o )
22		nnpredcl 4660	. . . . . . . . 9 |- ( N e. om -> U. N e. om )
23	22	adantr 276	. . . . . . . 8 |- ( ( N e. om /\ X e. ℕ∞ ) -> U. N e. om )
24	21, 23	ffvelcdmd 5701	. . . . . . 7 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. 2o )
25		df2o3 6497	. . . . . . 7 |- 2o = { (/) , 1o }
26	24, 25	eleqtrdi 2289	. . . . . 6 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. { (/) , 1o } )
27		elpri 3646	. . . . . 6 |- ( ( X ` U. N ) e. { (/) , 1o } -> ( ( X ` U. N ) = (/) \/ ( X ` U. N ) = 1o ) )
28	26, 27	syl 14	. . . . 5 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( ( X ` U. N ) = (/) \/ ( X ` U. N ) = 1o ) )
29	28	ad2antrr 488	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> ( ( X ` U. N ) = (/) \/ ( X ` U. N ) = 1o ) )
30	13, 19, 29	mpjaodan 799	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
31		nndceq0 4655	. . . . 5 |- ( N e. om -> DECID N = (/) )
32		exmiddc 837	. . . . 5 |- (DECID N = (/) -> ( N = (/) \/ -. N = (/) ) )
33	31, 32	syl 14	. . . 4 |- ( N e. om -> ( N = (/) \/ -. N = (/) ) )
34	33	ad2antrr 488	. . 3 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> ( N = (/) \/ -. N = (/) ) )
35	5, 30, 34	mpjaodan 799	. 2 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
36		1n0 6499	. . . . . 6 |- 1o =/= (/)
37	36	neii 2369	. . . . 5 |- -. 1o = (/)
38		simpr 110	. . . . . . . 8 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) /\ ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) -> ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
39	38	fveq1d 5563	. . . . . . 7 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) /\ ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = ( X ` N ) )
40		eqid 2196	. . . . . . . . . 10 |- ( i e. om |-> if ( i e. N , 1o , (/) ) ) = ( i e. om |-> if ( i e. N , 1o , (/) ) )
41		eleq1 2259	. . . . . . . . . . 11 |- ( i = N -> ( i e. N <-> N e. N ) )
42	41	ifbid 3583	. . . . . . . . . 10 |- ( i = N -> if ( i e. N , 1o , (/) ) = if ( N e. N , 1o , (/) ) )
43		id 19	. . . . . . . . . 10 |- ( N e. om -> N e. om )
44		nnord 4649	. . . . . . . . . . . . 13 |- ( N e. om -> Ord N )
45		ordirr 4579	. . . . . . . . . . . . 13 |- ( Ord N -> -. N e. N )
46	44, 45	syl 14	. . . . . . . . . . . 12 |- ( N e. om -> -. N e. N )
47	46	iffalsed 3572	. . . . . . . . . . 11 |- ( N e. om -> if ( N e. N , 1o , (/) ) = (/) )
48		peano1 4631	. . . . . . . . . . 11 |- (/) e. om
49	47, 48	eqeltrdi 2287	. . . . . . . . . 10 |- ( N e. om -> if ( N e. N , 1o , (/) ) e. om )
50	40, 42, 43, 49	fvmptd3 5658	. . . . . . . . 9 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = if ( N e. N , 1o , (/) ) )
51	50, 47	eqtrd 2229	. . . . . . . 8 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = (/) )
52	51	ad3antrrr 492	. . . . . . 7 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) /\ ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = (/) )
53		simplr 528	. . . . . . 7 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) /\ ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) -> ( X ` N ) = 1o )
54	39, 52, 53	3eqtr3rd 2238	. . . . . 6 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) /\ ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) -> 1o = (/) )
55	54	ex 115	. . . . 5 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X -> 1o = (/) ) )
56	37, 55	mtoi 665	. . . 4 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
57	56	olcd 735	. . 3 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X \/ -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) )
58		df-dc 836	. . 3 |- (DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X <-> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X \/ -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X ) )
59	57, 58	sylibr 134	. 2 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
60		simpl 109	. . . . 5 |- ( ( N e. om /\ X e. ℕ∞ ) -> N e. om )
61	21, 60	ffvelcdmd 5701	. . . 4 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. 2o )
62	61, 25	eleqtrdi 2289	. . 3 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. { (/) , 1o } )
63		elpri 3646	. . 3 |- ( ( X ` N ) e. { (/) , 1o } -> ( ( X ` N ) = (/) \/ ( X ` N ) = 1o ) )
64	62, 63	syl 14	. 2 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( ( X ` N ) = (/) \/ ( X ` N ) = 1o ) )
65	35, 59, 64	mpjaodan 799	1 |- ( ( N e. om /\ X e. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   (/)c0 3451   ifcif 3562   {cpr 3624   U.cuni 3840   |-> cmpt 4095   Ord word 4398   omcom 4627   -->wf 5255   ` cfv 5259   1oc1o 6476   2oc2o 6477  ℕ_∞xnninf 7194

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-fv 5267  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1o 6483  df-2o 6484  df-map 6718  df-nninf 7195

This theorem is referenced by: (None)