Theorem nninfisol 7479

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 7526). (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 540	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> X e. ℕ∞ )
2		simplr 533	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> ( X ` N ) = (/) )
3		simplll 539	. . . 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 7476	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
6		simp-4r 548	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> X e. ℕ∞ )
7		simpllr 540	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = (/) ) -> ( X ` N ) = (/) )
8		simp-4l 547	. . . . 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 2427	. . . . . 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 7477	. . . 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 548	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> X e. ℕ∞ )
15		simpllr 540	. . . . 5 |- ( ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) /\ ( X ` U. N ) = 1o ) -> ( X ` N ) = (/) )
16		simp-4l 547	. . . . 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 7478	. . . 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 7468	. . . . . . . . 9 |- ( X e. ℕ∞ -> X : om --> 2o )
21	20	adantl 277	. . . . . . . 8 |- ( ( N e. om /\ X e. ℕ∞ ) -> X : om --> 2o )
22		nnpredcl 4772	. . . . . . . . 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 5847	. . . . . . 7 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. 2o )
25		df2o3 6706	. . . . . . 7 |- 2o = { (/) , 1o }
26	24, 25	eleqtrdi 2331	. . . . . 6 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. { (/) , 1o } )
27		elpri 3732	. . . . . 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 492	. . . 4 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> ( ( X ` U. N ) = (/) \/ ( X ` U. N ) = 1o ) )
30	13, 19, 29	mpjaodan 810	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
31		nndceq0 4767	. . . . 5 |- ( N e. om -> DECID N = (/) )
32		exmiddc 848	. . . . 5 |- (DECID N = (/) -> ( N = (/) \/ -. N = (/) ) )
33	31, 32	syl 14	. . . 4 |- ( N e. om -> ( N = (/) \/ -. N = (/) ) )
34	33	ad2antrr 492	. . 3 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> ( N = (/) \/ -. N = (/) ) )
35	5, 30, 34	mpjaodan 810	. 2 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
36		1n0 6709	. . . . . 6 |- 1o =/= (/)
37	36	neii 2422	. . . . 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 5700	. . . . . . 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 2238	. . . . . . . . . 10 |- ( i e. om |-> if ( i e. N , 1o , (/) ) ) = ( i e. om |-> if ( i e. N , 1o , (/) ) )
41		eleq1 2301	. . . . . . . . . . 11 |- ( i = N -> ( i e. N <-> N e. N ) )
42	41	ifbid 3662	. . . . . . . . . 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 4761	. . . . . . . . . . . . 13 |- ( N e. om -> Ord N )
45		ordirr 4691	. . . . . . . . . . . . 13 |- ( Ord N -> -. N e. N )
46	44, 45	syl 14	. . . . . . . . . . . 12 |- ( N e. om -> -. N e. N )
47	46	iffalsed 3650	. . . . . . . . . . 11 |- ( N e. om -> if ( N e. N , 1o , (/) ) = (/) )
48		peano1 4743	. . . . . . . . . . 11 |- (/) e. om
49	47, 48	eqeltrdi 2329	. . . . . . . . . 10 |- ( N e. om -> if ( N e. N , 1o , (/) ) e. om )
50	40, 42, 43, 49	fvmptd3 5802	. . . . . . . . 9 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = if ( N e. N , 1o , (/) ) )
51	50, 47	eqtrd 2271	. . . . . . . 8 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = (/) )
52	51	ad3antrrr 496	. . . . . . 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 533	. . . . . . 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 2280	. . . . . 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 674	. . . 4 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
57	56	olcd 746	. . 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 847	. . 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 5847	. . . 4 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. 2o )
62	61, 25	eleqtrdi 2331	. . 3 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. { (/) , 1o } )
63		elpri 3732	. . 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 810	1 |- ( ( N e. om /\ X e. ℕ∞ ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   = wceq 1402   e. wcel 2209   =/= wne 2420   (/)c0 3520   ifcif 3638   {cpr 3710   U.cuni 3935   |-> cmpt 4193   Ord word 4509   omcom 4739   -->wf 5376   ` cfv 5380   1oc1o 6684   2oc2o 6685  ℕ_∞xnninf 7465

This proof depends on 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 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1o 6691  df-2o 6692  df-map 6928  df-nninf 7466

This theorem is used by: (None)