Theorem nninfisol 7426

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 7473). (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 536	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> X e. ℕ∞ )
2		simplr 529	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> ( X ` N ) = (/) )
3		simplll 535	. . . 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 7423	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
6		simp-4r 544	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> X e. ℕ∞ )
7		simpllr 536	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> ( X ` N ) = (/) )
8		simp-4l 543	. . . . 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 2421	. . . . . 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 7424	. . . 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 544	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> X e. ℕ∞ )
15		simpllr 536	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> ( X ` N ) = (/) )
16		simp-4l 543	. . . . 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 7425	. . . 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 7415	. . . . . . . . 9 |- ( X e. ℕ∞ -> X : om --> 2o )
21	20	adantl 277	. . . . . . . 8 |- ( ( N e. om /\ X e. ℕ∞ ) -> X : om --> 2o )
22		nnpredcl 4747	. . . . . . . . 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 5815	. . . . . . 7 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. 2o )
25		df2o3 6664	. . . . . . 7 |- 2o = { (/) , 1o }
26	24, 25	eleqtrdi 2327	. . . . . 6 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. { (/) , 1o } )
27		elpri 3714	. . . . . 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 806	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
31		nndceq0 4742	. . . . 5 |- ( N e. om -> DECID N = (/) )
32		exmiddc 844	. . . . 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 806	. 2 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
36		1n0 6667	. . . . . 6 |- 1o =/= (/)
37	36	neii 2416	. . . . 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 5674	. . . . . . 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 2234	. . . . . . . . . 10 |- ( i e. om |-> if ( i e. N , 1o , (/) ) ) = ( i e. om |-> if ( i e. N , 1o , (/) ) )
41		eleq1 2297	. . . . . . . . . . 11 |- ( i = N -> ( i e. N <-> N e. N ) )
42	41	ifbid 3646	. . . . . . . . . 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 4736	. . . . . . . . . . . . 13 |- ( N e. om -> Ord N )
45		ordirr 4666	. . . . . . . . . . . . 13 |- ( Ord N -> -. N e. N )
46	44, 45	syl 14	. . . . . . . . . . . 12 |- ( N e. om -> -. N e. N )
47	46	iffalsed 3634	. . . . . . . . . . 11 |- ( N e. om -> if ( N e. N , 1o , (/) ) = (/) )
48		peano1 4718	. . . . . . . . . . 11 |- (/) e. om
49	47, 48	eqeltrdi 2325	. . . . . . . . . 10 |- ( N e. om -> if ( N e. N , 1o , (/) ) e. om )
50	40, 42, 43, 49	fvmptd3 5773	. . . . . . . . 9 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = if ( N e. N , 1o , (/) ) )
51	50, 47	eqtrd 2267	. . . . . . . 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 529	. . . . . . 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 2276	. . . . . 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 670	. . . 4 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
57	56	olcd 742	. . 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 843	. . 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 5815	. . . 4 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. 2o )
62	61, 25	eleqtrdi 2327	. . 3 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. { (/) , 1o } )
63		elpri 3714	. . 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 806	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 716  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   (/)c0 3510   ifcif 3622   {cpr 3692   U.cuni 3916   |-> cmpt 4173   Ord word 4485   omcom 4714   -->wf 5350   ` cfv 5354   1oc1o 6642   2oc2o 6643  ℕ_∞xnninf 7412

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1o 6649  df-2o 6650  df-map 6886  df-nninf 7413

This theorem is referenced by: (None)