Theorem xnn0nnen 10508

Description: The set of extended nonnegative integers is equinumerous to the set of natural numbers. (Contributed by Jim Kingdon, 14-Jul-2025.)

Assertion

Ref	Expression
xnn0nnen	|- NN0* ~~ NN

Proof of Theorem xnn0nnen

Step	Hyp	Ref	Expression
1		fnresi 5371	. . . . . . . 8 |- ( _I |` NN0 ) Fn NN0
2		pnfex 8073	. . . . . . . . 9 |- +oo e. _V
3		neg1z 9349	. . . . . . . . . 10 |- -u 1 e. ZZ
4	3	elexi 2772	. . . . . . . . 9 |- -u 1 e. _V
5	2, 4	fnsn 5308	. . . . . . . 8 |- { <. +oo , -u 1 >. } Fn { +oo }
6	1, 5	pm3.2i 272	. . . . . . 7 |- ( ( _I |` NN0 ) Fn NN0 /\ { <. +oo , -u 1 >. } Fn { +oo } )
7		disj 3495	. . . . . . . 8 |- ( ( NN0 i^i { +oo } ) = (/) <-> A. x e. NN0 -. x e. { +oo } )
8		nn0nepnf 9311	. . . . . . . . 9 |- ( x e. NN0 -> x =/= +oo )
9		nelsn 3653	. . . . . . . . 9 |- ( x =/= +oo -> -. x e. { +oo } )
10	8, 9	syl 14	. . . . . . . 8 |- ( x e. NN0 -> -. x e. { +oo } )
11	7, 10	mprgbir 2552	. . . . . . 7 |- ( NN0 i^i { +oo } ) = (/)
12		fnun 5360	. . . . . . 7 |- ( ( ( ( _I |` NN0 ) Fn NN0 /\ { <. +oo , -u 1 >. } Fn { +oo } ) /\ ( NN0 i^i { +oo } ) = (/) ) -> ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) Fn ( NN0 u. { +oo } ) )
13	6, 11, 12	mp2an 426	. . . . . 6 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) Fn ( NN0 u. { +oo } )
14		uncom 3303	. . . . . . 7 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) = ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) )
15		df-xnn0 9304	. . . . . . . 8 |- NN0* = ( NN0 u. { +oo } )
16	15	eqcomi 2197	. . . . . . 7 |- ( NN0 u. { +oo } ) = NN0*
17		fneq12 5347	. . . . . . 7 |- ( ( ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) = ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) /\ ( NN0 u. { +oo } ) = NN0* ) -> ( ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) Fn ( NN0 u. { +oo } ) <-> ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn NN0* ) )
18	14, 16, 17	mp2an 426	. . . . . 6 |- ( ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) Fn ( NN0 u. { +oo } ) <-> ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn NN0* )
19	13, 18	mpbi 145	. . . . 5 |- ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn NN0*
20	4, 2	fnsn 5308	. . . . . . . . . 10 |- { <. -u 1 , +oo >. } Fn { -u 1 }
21	20, 1	pm3.2i 272	. . . . . . . . 9 |- ( { <. -u 1 , +oo >. } Fn { -u 1 } /\ ( _I |` NN0 ) Fn NN0 )
22		disj 3495	. . . . . . . . . 10 |- ( ( { -u 1 } i^i NN0 ) = (/) <-> A. x e. { -u 1 } -. x e. NN0 )
23		neg1lt0 9090	. . . . . . . . . . . 12 |- -u 1 < 0
24		nn0nlt0 9266	. . . . . . . . . . . 12 |- ( -u 1 e. NN0 -> -. -u 1 < 0 )
25	23, 24	mt2 641	. . . . . . . . . . 11 |- -. -u 1 e. NN0
26		elsni 3636	. . . . . . . . . . . 12 |- ( x e. { -u 1 } -> x = -u 1 )
27	26	eleq1d 2262	. . . . . . . . . . 11 |- ( x e. { -u 1 } -> ( x e. NN0 <-> -u 1 e. NN0 ) )
28	25, 27	mtbiri 676	. . . . . . . . . 10 |- ( x e. { -u 1 } -> -. x e. NN0 )
29	22, 28	mprgbir 2552	. . . . . . . . 9 |- ( { -u 1 } i^i NN0 ) = (/)
30		fnun 5360	. . . . . . . . 9 |- ( ( ( { <. -u 1 , +oo >. } Fn { -u 1 } /\ ( _I |` NN0 ) Fn NN0 ) /\ ( { -u 1 } i^i NN0 ) = (/) ) -> ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 ) )
31	21, 29, 30	mp2an 426	. . . . . . . 8 |- ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 )
32		cnvun 5071	. . . . . . . . . 10 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) )
33	2, 4	cnvsn 5148	. . . . . . . . . . 11 |- `' { <. +oo , -u 1 >. } = { <. -u 1 , +oo >. }
34		cnvresid 5328	. . . . . . . . . . 11 |- `' ( _I |` NN0 ) = ( _I |` NN0 )
35	33, 34	uneq12i 3311	. . . . . . . . . 10 |- ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
36	32, 35	eqtri 2214	. . . . . . . . 9 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
37	36	fneq1i 5348	. . . . . . . 8 |- ( `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 ) <-> ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 ) )
38	31, 37	mpbir 146	. . . . . . 7 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 )
39		fzosn 10272	. . . . . . . . . . 11 |- ( -u 1 e. ZZ -> ( -u 1..^ ( -u 1 + 1 ) ) = { -u 1 } )
40	3, 39	ax-mp 5	. . . . . . . . . 10 |- ( -u 1..^ ( -u 1 + 1 ) ) = { -u 1 }
41		ax-1cn 7965	. . . . . . . . . . . . 13 |- 1 e. CC
42	41, 41	negsubdii 8304	. . . . . . . . . . . 12 |- -u ( 1 - 1 ) = ( -u 1 + 1 )
43		1m1e0 9051	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
44	41, 41	subcli 8295	. . . . . . . . . . . . . 14 |- ( 1 - 1 ) e. CC
45		negeq0 8273	. . . . . . . . . . . . . 14 |- ( ( 1 - 1 ) e. CC -> ( ( 1 - 1 ) = 0 <-> -u ( 1 - 1 ) = 0 ) )
46	44, 45	ax-mp 5	. . . . . . . . . . . . 13 |- ( ( 1 - 1 ) = 0 <-> -u ( 1 - 1 ) = 0 )
47	43, 46	mpbi 145	. . . . . . . . . . . 12 |- -u ( 1 - 1 ) = 0
48	42, 47	eqtr3i 2216	. . . . . . . . . . 11 |- ( -u 1 + 1 ) = 0
49	48	oveq2i 5929	. . . . . . . . . 10 |- ( -u 1..^ ( -u 1 + 1 ) ) = ( -u 1..^ 0 )
50	40, 49	eqtr3i 2216	. . . . . . . . 9 |- { -u 1 } = ( -u 1..^ 0 )
51		nn0uz 9627	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	uneq12i 3311	. . . . . . . 8 |- ( { -u 1 } u. NN0 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
53	52	fneq2i 5349	. . . . . . 7 |- ( `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 ) <-> `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) ) )
54	38, 53	mpbi 145	. . . . . 6 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
55		0z 9328	. . . . . . . . 9 |- 0 e. ZZ
56		neg1rr 9088	. . . . . . . . . 10 |- -u 1 e. RR
57		0re 8019	. . . . . . . . . 10 |- 0 e. RR
58	56, 57, 23	ltleii 8122	. . . . . . . . 9 |- -u 1 <_ 0
59		eluz2 9598	. . . . . . . . 9 |- ( 0 e. ( ZZ>= ` -u 1 ) <-> ( -u 1 e. ZZ /\ 0 e. ZZ /\ -u 1 <_ 0 ) )
60	3, 55, 58, 59	mpbir3an 1181	. . . . . . . 8 |- 0 e. ( ZZ>= ` -u 1 )
61		fzouzsplit 10246	. . . . . . . 8 |- ( 0 e. ( ZZ>= ` -u 1 ) -> ( ZZ>= ` -u 1 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) ) )
62	60, 61	ax-mp 5	. . . . . . 7 |- ( ZZ>= ` -u 1 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
63	62	fneq2i 5349	. . . . . 6 |- ( `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ZZ>= ` -u 1 ) <-> `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) ) )
64	54, 63	mpbir 146	. . . . 5 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ZZ>= ` -u 1 )
65	19, 64	pm3.2i 272	. . . 4 |- ( ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn NN0* /\ `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ZZ>= ` -u 1 ) )
66		dff1o4 5508	. . . 4 |- ( ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) :NN0* -1-1-onto-> ( ZZ>= ` -u 1 ) <-> ( ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn NN0* /\ `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) Fn ( ZZ>= ` -u 1 ) ) )
67	65, 66	mpbir 146	. . 3 |- ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) :NN0* -1-1-onto-> ( ZZ>= ` -u 1 )
68		nn0ex 9246	. . . . . 6 |- NN0 e. _V
69	2	snex 4214	. . . . . 6 |- { +oo } e. _V
70	68, 69	unex 4472	. . . . 5 |- ( NN0 u. { +oo } ) e. _V
71	15, 70	eqeltri 2266	. . . 4 |- NN0* e. _V
72	71	f1oen 6813	. . 3 |- ( ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) :NN0* -1-1-onto-> ( ZZ>= ` -u 1 ) -> NN0* ~~ ( ZZ>= ` -u 1 ) )
73	67, 72	ax-mp 5	. 2 |- NN0* ~~ ( ZZ>= ` -u 1 )
74		uzennn 10507	. . 3 |- ( -u 1 e. ZZ -> ( ZZ>= ` -u 1 ) ~~ NN )
75	3, 74	ax-mp 5	. 2 |- ( ZZ>= ` -u 1 ) ~~ NN
76	73, 75	entri 6840	1 |- NN0* ~~ NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   =/= wne 2364   _Vcvv 2760   u. cun 3151   i^i cin 3152   (/)c0 3446   {csn 3618   <.cop 3621   class class class wbr 4029   _I cid 4319   `'ccnv 4658   |` cres 4661   Fn wfn 5249   -1-1-onto->wf1o 5253   ` cfv 5254  (class class class)co 5918   ~~ cen 6792   CCcc 7870   0cc0 7872   1c1 7873   + caddc 7875   +oocpnf 8051   < clt 8054   <_ cle 8055   - cmin 8190   -ucneg 8191   NNcn 8982   NN0cn0 9240  NN0*cxnn0 9303   ZZcz 9317   ZZ>=cuz 9592  ..^cfzo 10208

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-er 6587  df-en 6795  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-inn 8983  df-n0 9241  df-xnn0 9304  df-z 9318  df-uz 9593  df-fz 10075  df-fzo 10209

This theorem is referenced by:  nninfct  12178