Theorem xnn0nnen 10874

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 5501	. . . . . . . 8 |- ( _I |` NN0 ) Fn NN0
2		pnfex 8379	. . . . . . . . 9 |- +oo e. _V
3		neg1z 9676	. . . . . . . . . 10 |- -u 1 e. ZZ
4	3	elexi 2834	. . . . . . . . 9 |- -u 1 e. _V
5	2, 4	fnsn 5435	. . . . . . . 8 |- { <. +oo , -u 1 >. } Fn { +oo }
6	1, 5	pm3.2i 272	. . . . . . 7 |- ( ( _I |` NN0 ) Fn NN0 /\ { <. +oo , -u 1 >. } Fn { +oo } )
7		disj 3573	. . . . . . . 8 |- ( ( NN0 i^i { +oo } ) = (/) <-> A. x e. NN0 -. x e. { +oo } )
8		nn0nepnf 9638	. . . . . . . . 9 |- ( x e. NN0 -> x =/= +oo )
9		nelsn 3744	. . . . . . . . 9 |- ( x =/= +oo -> -. x e. { +oo } )
10	8, 9	syl 14	. . . . . . . 8 |- ( x e. NN0 -> -. x e. { +oo } )
11	7, 10	mprgbir 2608	. . . . . . 7 |- ( NN0 i^i { +oo } ) = (/)
12		fnun 5489	. . . . . . 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 430	. . . . . 6 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) Fn ( NN0 u. { +oo } )
14		uncom 3373	. . . . . . 7 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) = ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) )
15		df-xnn0 9631	. . . . . . . 8 |- NN0* = ( NN0 u. { +oo } )
16	15	eqcomi 2242	. . . . . . 7 |- ( NN0 u. { +oo } ) = NN0*
17		fneq12 5474	. . . . . . 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 430	. . . . . 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 5435	. . . . . . . . . 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 3573	. . . . . . . . . 10 |- ( ( { -u 1 } i^i NN0 ) = (/) <-> A. x e. { -u 1 } -. x e. NN0 )
23		neg1lt0 9412	. . . . . . . . . . . 12 |- -u 1 < 0
24		nn0nlt0 9589	. . . . . . . . . . . 12 |- ( -u 1 e. NN0 -> -. -u 1 < 0 )
25	23, 24	mt2 649	. . . . . . . . . . 11 |- -. -u 1 e. NN0
26		elsni 3727	. . . . . . . . . . . 12 |- ( x e. { -u 1 } -> x = -u 1 )
27	26	eleq1d 2307	. . . . . . . . . . 11 |- ( x e. { -u 1 } -> ( x e. NN0 <-> -u 1 e. NN0 ) )
28	25, 27	mtbiri 686	. . . . . . . . . 10 |- ( x e. { -u 1 } -> -. x e. NN0 )
29	22, 28	mprgbir 2608	. . . . . . . . 9 |- ( { -u 1 } i^i NN0 ) = (/)
30		fnun 5489	. . . . . . . . 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 430	. . . . . . . 8 |- ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) ) Fn ( { -u 1 } u. NN0 )
32		cnvun 5193	. . . . . . . . . 10 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) )
33	2, 4	cnvsn 5270	. . . . . . . . . . 11 |- `' { <. +oo , -u 1 >. } = { <. -u 1 , +oo >. }
34		cnvresid 5455	. . . . . . . . . . 11 |- `' ( _I |` NN0 ) = ( _I |` NN0 )
35	33, 34	uneq12i 3381	. . . . . . . . . 10 |- ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
36	32, 35	eqtri 2259	. . . . . . . . 9 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
37	36	fneq1i 5475	. . . . . . . 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 10623	. . . . . . . . . . 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 8272	. . . . . . . . . . . . 13 |- 1 e. CC
42	41, 41	negsubdii 8611	. . . . . . . . . . . 12 |- -u ( 1 - 1 ) = ( -u 1 + 1 )
43		1m1e0 9373	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
44	41, 41	subcli 8602	. . . . . . . . . . . . . 14 |- ( 1 - 1 ) e. CC
45		negeq0 8580	. . . . . . . . . . . . . 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 2261	. . . . . . . . . . 11 |- ( -u 1 + 1 ) = 0
49	48	oveq2i 6096	. . . . . . . . . 10 |- ( -u 1..^ ( -u 1 + 1 ) ) = ( -u 1..^ 0 )
50	40, 49	eqtr3i 2261	. . . . . . . . 9 |- { -u 1 } = ( -u 1..^ 0 )
51		nn0uz 9957	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	uneq12i 3381	. . . . . . . 8 |- ( { -u 1 } u. NN0 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
53	52	fneq2i 5476	. . . . . . 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 9655	. . . . . . . . 9 |- 0 e. ZZ
56		neg1rr 9410	. . . . . . . . . 10 |- -u 1 e. RR
57		0re 8326	. . . . . . . . . 10 |- 0 e. RR
58	56, 57, 23	ltleii 8428	. . . . . . . . 9 |- -u 1 <_ 0
59		eluz2 9927	. . . . . . . . 9 |- ( 0 e. ( ZZ>= ` -u 1 ) <-> ( -u 1 e. ZZ /\ 0 e. ZZ /\ -u 1 <_ 0 ) )
60	3, 55, 58, 59	mpbir3an 1210	. . . . . . . 8 |- 0 e. ( ZZ>= ` -u 1 )
61		fzouzsplit 10588	. . . . . . . 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 5476	. . . . . 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 5647	. . . 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 9569	. . . . . 6 |- NN0 e. _V
69	2	snex 4322	. . . . . 6 |- { +oo } e. _V
70	68, 69	unex 4587	. . . . 5 |- ( NN0 u. { +oo } ) e. _V
71	15, 70	eqeltri 2311	. . . 4 |- NN0* e. _V
72	71	f1oen 7045	. . 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 10873	. . 3 |- ( -u 1 e. ZZ -> ( ZZ>= ` -u 1 ) ~~ NN )
75	3, 74	ax-mp 5	. 2 |- ( ZZ>= ` -u 1 ) ~~ NN
76	73, 75	entri 7073	1 |- NN0* ~~ NN

This proof depends on syntax axioms:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   =/= wne 2420   _Vcvv 2821   u. cun 3218   i^i cin 3219   (/)c0 3520   {csn 3709   <.cop 3712   class class class wbr 4130   _I cid 4433   `'ccnv 4773   |` cres 4776   Fn wfn 5372   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   ~~ cen 7020   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   +oocpnf 8357   < clt 8360   <_ cle 8361   - cmin 8497   -ucneg 8498   NNcn 9304   NN0cn0 9563  NN0*cxnn0 9630   ZZcz 9644   ZZ>=cuz 9921  ..^cfzo 10549

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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  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-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  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-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-er 6807  df-en 7023  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-xnn0 9631  df-z 9645  df-uz 9922  df-fz 10412  df-fzo 10550

This theorem is used by:  nninfct  12818