Theorem xnn0nnen 10745

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 5457	. . . . . . . 8 |- ( _I |` NN0 ) Fn NN0
2		pnfex 8275	. . . . . . . . 9 |- +oo e. _V
3		neg1z 9555	. . . . . . . . . 10 |- -u 1 e. ZZ
4	3	elexi 2816	. . . . . . . . 9 |- -u 1 e. _V
5	2, 4	fnsn 5391	. . . . . . . 8 |- { <. +oo , -u 1 >. } Fn { +oo }
6	1, 5	pm3.2i 272	. . . . . . 7 |- ( ( _I |` NN0 ) Fn NN0 /\ { <. +oo , -u 1 >. } Fn { +oo } )
7		disj 3545	. . . . . . . 8 |- ( ( NN0 i^i { +oo } ) = (/) <-> A. x e. NN0 -. x e. { +oo } )
8		nn0nepnf 9517	. . . . . . . . 9 |- ( x e. NN0 -> x =/= +oo )
9		nelsn 3708	. . . . . . . . 9 |- ( x =/= +oo -> -. x e. { +oo } )
10	8, 9	syl 14	. . . . . . . 8 |- ( x e. NN0 -> -. x e. { +oo } )
11	7, 10	mprgbir 2591	. . . . . . 7 |- ( NN0 i^i { +oo } ) = (/)
12		fnun 5445	. . . . . . 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 3353	. . . . . . 7 |- ( ( _I |` NN0 ) u. { <. +oo , -u 1 >. } ) = ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) )
15		df-xnn0 9510	. . . . . . . 8 |- NN0* = ( NN0 u. { +oo } )
16	15	eqcomi 2235	. . . . . . 7 |- ( NN0 u. { +oo } ) = NN0*
17		fneq12 5430	. . . . . . 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 5391	. . . . . . . . . 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 3545	. . . . . . . . . 10 |- ( ( { -u 1 } i^i NN0 ) = (/) <-> A. x e. { -u 1 } -. x e. NN0 )
23		neg1lt0 9293	. . . . . . . . . . . 12 |- -u 1 < 0
24		nn0nlt0 9470	. . . . . . . . . . . 12 |- ( -u 1 e. NN0 -> -. -u 1 < 0 )
25	23, 24	mt2 645	. . . . . . . . . . 11 |- -. -u 1 e. NN0
26		elsni 3691	. . . . . . . . . . . 12 |- ( x e. { -u 1 } -> x = -u 1 )
27	26	eleq1d 2300	. . . . . . . . . . 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 2591	. . . . . . . . 9 |- ( { -u 1 } i^i NN0 ) = (/)
30		fnun 5445	. . . . . . . . 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 5149	. . . . . . . . . 10 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) )
33	2, 4	cnvsn 5226	. . . . . . . . . . 11 |- `' { <. +oo , -u 1 >. } = { <. -u 1 , +oo >. }
34		cnvresid 5411	. . . . . . . . . . 11 |- `' ( _I |` NN0 ) = ( _I |` NN0 )
35	33, 34	uneq12i 3361	. . . . . . . . . 10 |- ( `' { <. +oo , -u 1 >. } u. `' ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
36	32, 35	eqtri 2252	. . . . . . . . 9 |- `' ( { <. +oo , -u 1 >. } u. ( _I |` NN0 ) ) = ( { <. -u 1 , +oo >. } u. ( _I |` NN0 ) )
37	36	fneq1i 5431	. . . . . . . 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 10496	. . . . . . . . . . 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 8168	. . . . . . . . . . . . 13 |- 1 e. CC
42	41, 41	negsubdii 8506	. . . . . . . . . . . 12 |- -u ( 1 - 1 ) = ( -u 1 + 1 )
43		1m1e0 9254	. . . . . . . . . . . . 13 |- ( 1 - 1 ) = 0
44	41, 41	subcli 8497	. . . . . . . . . . . . . 14 |- ( 1 - 1 ) e. CC
45		negeq0 8475	. . . . . . . . . . . . . 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 2254	. . . . . . . . . . 11 |- ( -u 1 + 1 ) = 0
49	48	oveq2i 6039	. . . . . . . . . 10 |- ( -u 1..^ ( -u 1 + 1 ) ) = ( -u 1..^ 0 )
50	40, 49	eqtr3i 2254	. . . . . . . . 9 |- { -u 1 } = ( -u 1..^ 0 )
51		nn0uz 9835	. . . . . . . . 9 |- NN0 = ( ZZ>= ` 0 )
52	50, 51	uneq12i 3361	. . . . . . . 8 |- ( { -u 1 } u. NN0 ) = ( ( -u 1..^ 0 ) u. ( ZZ>= ` 0 ) )
53	52	fneq2i 5432	. . . . . . 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 9534	. . . . . . . . 9 |- 0 e. ZZ
56		neg1rr 9291	. . . . . . . . . 10 |- -u 1 e. RR
57		0re 8222	. . . . . . . . . 10 |- 0 e. RR
58	56, 57, 23	ltleii 8324	. . . . . . . . 9 |- -u 1 <_ 0
59		eluz2 9805	. . . . . . . . 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 10461	. . . . . . . 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 5432	. . . . . 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 5600	. . . 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 9450	. . . . . 6 |- NN0 e. _V
69	2	snex 4281	. . . . . 6 |- { +oo } e. _V
70	68, 69	unex 4544	. . . . 5 |- ( NN0 u. { +oo } ) e. _V
71	15, 70	eqeltri 2304	. . . 4 |- NN0* e. _V
72	71	f1oen 6975	. . 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 10744	. . 3 |- ( -u 1 e. ZZ -> ( ZZ>= ` -u 1 ) ~~ NN )
75	3, 74	ax-mp 5	. 2 |- ( ZZ>= ` -u 1 ) ~~ NN
76	73, 75	entri 7003	1 |- NN0* ~~ NN

Syntax hints:   -. wn 3   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   u. cun 3199   i^i cin 3200   (/)c0 3496   {csn 3673   <.cop 3676   class class class wbr 4093   _I cid 4391   `'ccnv 4730   |` cres 4733   Fn wfn 5328   -1-1-onto->wf1o 5332   ` cfv 5333  (class class class)co 6028   ~~ cen 6950   CCcc 8073   0cc0 8075   1c1 8076   + caddc 8078   +oocpnf 8253   < clt 8256   <_ cle 8257   - cmin 8392   -ucneg 8393   NNcn 9185   NN0cn0 9444  NN0*cxnn0 9509   ZZcz 9523   ZZ>=cuz 9799  ..^cfzo 10422

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-er 6745  df-en 6953  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-xnn0 9510  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423

This theorem is referenced by:  nninfct  12675