Theorem rp-isfinite6 36076

Description: A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some n e. NN. (Contributed by Richard Penner, 10-Mar-2020.)

Assertion

Ref	Expression
rp-isfinite6	|- ( A e. Fin <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )

Distinct variable group:   A, n

Proof of Theorem rp-isfinite6

Step	Hyp	Ref	Expression
1		exmid 416	. . . 4 |- ( A = (/) \/ -. A = (/) )
2	1	biantrur 508	. . 3 |- ( A e. Fin <-> ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) )
3		andir 876	. . 3 |- ( ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
4	2, 3	bitri 252	. 2 |- ( A e. Fin <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
5		simpl 458	. . . 4 |- ( ( A = (/) /\ A e. Fin ) -> A = (/) )
6		0fin 7752	. . . . . 6 |- (/) e. Fin
7		eleq1a 2501	. . . . . 6 |- ( (/) e. Fin -> ( A = (/) -> A e. Fin ) )
8	6, 7	ax-mp 5	. . . . 5 |- ( A = (/) -> A e. Fin )
9	8	ancli 553	. . . 4 |- ( A = (/) -> ( A = (/) /\ A e. Fin ) )
10	5, 9	impbii 190	. . 3 |- ( ( A = (/) /\ A e. Fin ) <-> A = (/) )
11		rp-isfinite5 36075	. . . . . 6 |- ( A e. Fin <-> E. n e. NN0 ( 1 ... n ) ~~ A )
12		df-rex 2720	. . . . . 6 |- ( E. n e. NN0 ( 1 ... n ) ~~ A <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
13	11, 12	bitri 252	. . . . 5 |- ( A e. Fin <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
14	13	anbi2i 698	. . . 4 |- ( ( -. A = (/) /\ A e. Fin ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
15		df-rex 2720	. . . . 5 |- ( E. n e. NN ( 1 ... n ) ~~ A <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
16		en0 7586	. . . . . . . . . . . . . . 15 |- ( A ~~ (/) <-> A = (/) )
17	16	bicomi 205	. . . . . . . . . . . . . 14 |- ( A = (/) <-> A ~~ (/) )
18		ensymb 7571	. . . . . . . . . . . . . 14 |- ( A ~~ (/) <-> (/) ~~ A )
19	17, 18	bitri 252	. . . . . . . . . . . . 13 |- ( A = (/) <-> (/) ~~ A )
20	19	notbii 297	. . . . . . . . . . . 12 |- ( -. A = (/) <-> -. (/) ~~ A )
21		elnn0 10822	. . . . . . . . . . . . . 14 |- ( n e. NN0 <-> ( n e. NN \/ n = 0 ) )
22	21	anbi1i 699	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) )
23		andir 876	. . . . . . . . . . . . 13 |- ( ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
24	22, 23	bitri 252	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
25	20, 24	anbi12i 701	. . . . . . . . . . 11 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. (/) ~~ A /\ ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
26		andi 875	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
27	25, 26	bitri 252	. . . . . . . . . 10 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
28		3anass 986	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) )
29		3anass 986	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
30	28, 29	orbi12i 523	. . . . . . . . . . 11 |- ( ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
31	30	bicomi 205	. . . . . . . . . 10 |- ( ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) <-> ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) )
32	27, 31	sylbb 200	. . . . . . . . 9 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) )
33		simp2 1006	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) -> n e. NN )
34		oveq2 6257	. . . . . . . . . . . 12 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
35		fz10 11771	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
36	34, 35	syl6eq 2478	. . . . . . . . . . 11 |- ( n = 0 -> ( 1 ... n ) = (/) )
37		simp2 1006	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) = (/) )
38		simp3 1007	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) ~~ A )
39	37, 38	eqbrtrrd 4389	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> (/) ~~ A )
40		simp1 1005	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
41	39, 40	pm2.21dd 177	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> n e. NN )
42	36, 41	syl3an2 1298	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) -> n e. NN )
43	33, 42	jaoi 380	. . . . . . . . 9 |- ( ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
44	32, 43	syl 17	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
45		simprr 764	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( 1 ... n ) ~~ A )
46	44, 45	jca 534	. . . . . . 7 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
47		nngt0 10589	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
48		hash0 12498	. . . . . . . . . . . . 13 |- ( # ` (/) ) = 0
49	48	a1i 11	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` (/) ) = 0 )
50		nnnn0 10827	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
51		hashfz1 12479	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( # ` ( 1 ... n ) ) = n )
52	50, 51	syl 17	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` ( 1 ... n ) ) = n )
53	47, 49, 52	3brtr4d 4397	. . . . . . . . . . 11 |- ( n e. NN -> ( # ` (/) ) < ( # ` ( 1 ... n ) ) )
54		fzfi 12135	. . . . . . . . . . . 12 |- ( 1 ... n ) e. Fin
55		hashsdom 12510	. . . . . . . . . . . 12 |- ( ( (/) e. Fin /\ ( 1 ... n ) e. Fin ) -> ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) ) )
56	6, 54, 55	mp2an 676	. . . . . . . . . . 11 |- ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) )
57	53, 56	sylib 199	. . . . . . . . . 10 |- ( n e. NN -> (/) ~< ( 1 ... n ) )
58	57	anim1i 570	. . . . . . . . 9 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) )
59		sdomentr 7659	. . . . . . . . . . 11 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> (/) ~< A )
60		sdomnen 7552	. . . . . . . . . . 11 |- ( (/) ~< A -> -. (/) ~~ A )
61	59, 60	syl 17	. . . . . . . . . 10 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
62		ensymb 7571	. . . . . . . . . . . 12 |- ( (/) ~~ A <-> A ~~ (/) )
63	62, 16	bitri 252	. . . . . . . . . . 11 |- ( (/) ~~ A <-> A = (/) )
64	63	notbii 297	. . . . . . . . . 10 |- ( -. (/) ~~ A <-> -. A = (/) )
65	61, 64	sylib 199	. . . . . . . . 9 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
66	58, 65	syl 17	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
67	50	anim1i 570	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
68	66, 67	jca 534	. . . . . . 7 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
69	46, 68	impbii 190	. . . . . 6 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
70	69	exbii 1712	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
71		19.42v 1827	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
72	15, 70, 71	3bitr2ri 277	. . . 4 |- ( ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n e. NN ( 1 ... n ) ~~ A )
73	14, 72	bitri 252	. . 3 |- ( ( -. A = (/) /\ A e. Fin ) <-> E. n e. NN ( 1 ... n ) ~~ A )
74	10, 73	orbi12i 523	. 2 |- ( ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )
75	4, 74	bitri 252	1 |- ( A e. Fin <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   \/ wo 369   /\ wa 370   /\ w3a 982   = wceq 1437   E.wex 1657   e. wcel 1872   E.wrex 2715   (/)c0 3704   class class class wbr 4366   ` cfv 5544  (class class class)co 6249   ~~ cen 7521   ~< csdm 7523   Fincfn 7524   0cc0 9490   1c1 9491   < clt 9626   NNcn 10560   NN0cn0 10820   ...cfz 11735   #chash 12465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-int 4199  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-1o 7137  df-oadd 7141  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-fin 7528  df-card 8325  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-n0 10821  df-z 10889  df-uz 11111  df-fz 11736  df-hash 12466

This theorem is referenced by: (None)