Theorem nninfisol 7133

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 7130	. . 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 2354	. . . . . 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 7131	. . . 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 7132	. . . 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 7123	. . . . . . . . 9 |- ( X e. ℕ∞ -> X : om --> 2o )
21	20	adantl 277	. . . . . . . 8 |- ( ( N e. om /\ X e. ℕ∞ ) -> X : om --> 2o )
22		nnpredcl 4624	. . . . . . . . 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 5654	. . . . . . 7 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. 2o )
25		df2o3 6433	. . . . . . 7 |- 2o = { (/) , 1o }
26	24, 25	eleqtrdi 2270	. . . . . 6 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` U. N ) e. { (/) , 1o } )
27		elpri 3617	. . . . . 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 798	. . 3 |- ( ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) /\ -. N = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
31		nndceq0 4619	. . . . 5 |- ( N e. om -> DECID N = (/) )
32		exmiddc 836	. . . . 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 798	. 2 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = (/) ) -> DECID ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
36		1n0 6435	. . . . . 6 |- 1o =/= (/)
37	36	neii 2349	. . . . 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 5519	. . . . . . 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 2177	. . . . . . . . . 10 |- ( i e. om |-> if ( i e. N , 1o , (/) ) ) = ( i e. om |-> if ( i e. N , 1o , (/) ) )
41		eleq1 2240	. . . . . . . . . . 11 |- ( i = N -> ( i e. N <-> N e. N ) )
42	41	ifbid 3557	. . . . . . . . . 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 4613	. . . . . . . . . . . . 13 |- ( N e. om -> Ord N )
45		ordirr 4543	. . . . . . . . . . . . 13 |- ( Ord N -> -. N e. N )
46	44, 45	syl 14	. . . . . . . . . . . 12 |- ( N e. om -> -. N e. N )
47	46	iffalsed 3546	. . . . . . . . . . 11 |- ( N e. om -> if ( N e. N , 1o , (/) ) = (/) )
48		peano1 4595	. . . . . . . . . . 11 |- (/) e. om
49	47, 48	eqeltrdi 2268	. . . . . . . . . 10 |- ( N e. om -> if ( N e. N , 1o , (/) ) e. om )
50	40, 42, 43, 49	fvmptd3 5611	. . . . . . . . 9 |- ( N e. om -> ( ( i e. om |-> if ( i e. N , 1o , (/) ) ) ` N ) = if ( N e. N , 1o , (/) ) )
51	50, 47	eqtrd 2210	. . . . . . . 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 2219	. . . . . 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 664	. . . 4 |- ( ( ( N e. om /\ X e. ℕ∞ ) /\ ( X ` N ) = 1o ) -> -. ( i e. om |-> if ( i e. N , 1o , (/) ) ) = X )
57	56	olcd 734	. . 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 835	. . 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 5654	. . . 4 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. 2o )
62	61, 25	eleqtrdi 2270	. . 3 |- ( ( N e. om /\ X e. ℕ∞ ) -> ( X ` N ) e. { (/) , 1o } )
63		elpri 3617	. . 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 798	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 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   (/)c0 3424   ifcif 3536   {cpr 3595   U.cuni 3811   |-> cmpt 4066   Ord word 4364   omcom 4591   -->wf 5214   ` cfv 5218   1oc1o 6412   2oc2o 6413  ℕ_∞xnninf 7120

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1o 6419  df-2o 6420  df-map 6652  df-nninf 7121

This theorem is referenced by: (None)