Theorem xnn0nnen 10799

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 5476	. . . . . . . 8 |- ( _I |` NN0 ) Fn NN0
2		pnfex 8327	. . . . . . . . 9 |- +oo e. _V
3		neg1z 9609	. . . . . . . . . 10 |- -u 1 e. ZZ
4	3	elexi 2826	. . . . . . . . 9 |- -u 1 e. _V
5	2, 4	fnsn 5410	. . . . . . . 8 |- { <. +oo , -u 1 >. } Fn { +oo }
6	1, 5	pm3.2i 272	. . . . . . 7 |- ( ( _I |` NN0 ) Fn NN0 /\ { <. +oo , -u 1 >. } Fn { +oo } )
7		disj 3557	. . . . . . . 8 |- ( ( NN0 i^i { +oo } ) = (/) <-> A. x e. NN0 -. x e. { +oo } )
8		nn0nepnf 9571	. . . . . . . . 9 |- ( x e. NN0 -> x =/= +oo )
9		nelsn 3724	. . . . . . . . 9 |- ( x =/= +oo -> -. x e. { +oo } )
10	8, 9	syl 14	. . . . . . . 8 |- ( x e. NN0 -> -. x e. { +oo } )
11	7, 10	mprgbir 2600	. . . . . . 7 |- ( NN0 i^i { +oo } ) = (/)
12		fnun 5464	. . . . . . 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 3363	. . . . . . 7 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) = ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) )
15		df-xnn0 9564	. . . . . . . 8 |- NN0* = ( NN0 u. { +oo } )
16	15	eqcomi 2236	. . . . . . 7 |- ( NN0 u. { +oo } ) = NN0*
17		fneq12 5449	. . . . . . 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 5410	. . . . . . . . . 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 3557	. . . . . . . . . 10 |- ( ( { -u 1 } i^i NN0 ) = (/) <-> A. x e. { -u 1 } -. x e. NN0 )
23		neg1lt0 9345	. . . . . . . . . . . 12 |- -u 1 < 0
24		nn0nlt0 9522	. . . . . . . . . . . 12 |- ( -u 1 e. NN0 -> -. -u 1 < 0 )
25	23, 24	mt2 645	. . . . . . . . . . 11 |- -. -u 1 e. NN0
26		elsni 3707	. . . . . . . . . . . 12 |- ( x e. { -u 1 } -> x = -u 1 )
27	26	eleq1d 2301	. . . . . . . . . . 11 |- ( x e. { -u 1 } -> ( x e. NN0 <-> -u 1 e. NN0 ) )
28	25, 27	mtbiri 682	. . . . . . . . . 10 |- ( x e. { -u 1 } -> -. x e. NN0 )
29	22, 28	mprgbir 2600	. . . . . . . . 9 |- ( { -u 1 } i^i NN0 ) = (/)
30		fnun 5464	. . . . . . . . 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 5168	. . . . . . . . . 10 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) )
33	2, 4	cnvsn 5245	. . . . . . . . . . 11 |- `' { <. +oo , -u 1 >. } = { <. -u 1 , +oo >. }
34		cnvresid 5430	. . . . . . . . . . 11 |- `' ( _I |` NN0 ) = ( _I |` NN0 )
35	33, 34	uneq12i 3371	. . . . . . . . . 10 |- ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
36	32, 35	eqtri 2253	. . . . . . . . 9 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
37	36	fneq1i 5450	. . . . . . . 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 10550	. . . . . . . . . . 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 8220	. . . . . . . . . . . . 13 |- 1 e. CC
42	41, 41	negsubdii 8558	. . . . . . . . . . . 12 |- -u ( 1 - 1 ) = ( -u 1 + 1 )
43		1m1e0 9306	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
44	41, 41	subcli 8549	. . . . . . . . . . . . . 14 |- ( 1 - 1 ) e. CC
45		negeq0 8527	. . . . . . . . . . . . . 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 2255	. . . . . . . . . . 11 |- ( -u 1 + 1 ) = 0
49	48	oveq2i 6061	. . . . . . . . . 10 |- ( -u 1..^ ( -u 1 + 1 ) ) = ( -u 1..^ 0 )
50	40, 49	eqtr3i 2255	. . . . . . . . 9 |- { -u 1 } = ( -u 1..^ 0 )
51		nn0uz 9889	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	uneq12i 3371	. . . . . . . 8 |- ( { -u 1 } u. NN0 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
53	52	fneq2i 5451	. . . . . . 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 9588	. . . . . . . . 9 |- 0 e. ZZ
56		neg1rr 9343	. . . . . . . . . 10 |- -u 1 e. RR
57		0re 8274	. . . . . . . . . 10 |- 0 e. RR
58	56, 57, 23	ltleii 8376	. . . . . . . . 9 |- -u 1 <_ 0
59		eluz2 9859	. . . . . . . . 9 |- ( 0 e. ( ZZ>= ` -u 1 ) <-> ( -u 1 e. ZZ /\ 0 e. ZZ /\ -u 1 <_ 0 ) )
60	3, 55, 58, 59	mpbir3an 1206	. . . . . . . 8 |- 0 e. ( ZZ>= ` -u 1 )
61		fzouzsplit 10515	. . . . . . . 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 5451	. . . . . 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 5622	. . . 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 9502	. . . . . 6 |- NN0 e. _V
69	2	snex 4298	. . . . . 6 |- { +oo } e. _V
70	68, 69	unex 4562	. . . . 5 |- ( NN0 u. { +oo } ) e. _V
71	15, 70	eqeltri 2305	. . . 4 |- NN0* e. _V
72	71	f1oen 6998	. . 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 10798	. . 3 |- ( -u 1 e. ZZ -> ( ZZ>= ` -u 1 ) ~~ NN )
75	3, 74	ax-mp 5	. 2 |- ( ZZ>= ` -u 1 ) ~~ NN
76	73, 75	entri 7026	1 |- NN0* ~~ NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   =/= wne 2412   _Vcvv 2813   u. cun 3209   i^i cin 3210   (/)c0 3508   {csn 3689   <.cop 3692   class class class wbr 4109   _I cid 4409   `'ccnv 4748   |` cres 4751   Fn wfn 5347   -1-1-onto->wf1o 5351   ` cfv 5352  (class class class)co 6050   ~~ cen 6973   CCcc 8125   0cc0 8127   1c1 8128   + caddc 8130   +oocpnf 8305   < clt 8308   <_ cle 8309   - cmin 8444   -ucneg 8445   NNcn 9237   NN0cn0 9496  NN0*cxnn0 9563   ZZcz 9577   ZZ>=cuz 9853  ..^cfzo 10476

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-er 6767  df-en 6976  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-xnn0 9564  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477

This theorem is referenced by:  nninfct  12737