Theorem rp-isfinite6 37864

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 431	. . . 4 |- ( A = (/) \/ -. A = (/) )
2	1	biantrur 527	. . 3 |- ( A e. Fin <-> ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) )
3		andir 912	. . 3 |- ( ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
4	2, 3	bitri 264	. 2 |- ( A e. Fin <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
5		simpl 473	. . . 4 |- ( ( A = (/) /\ A e. Fin ) -> A = (/) )
6		0fin 8188	. . . . . 6 |- (/) e. Fin
7		eleq1a 2696	. . . . . 6 |- ( (/) e. Fin -> ( A = (/) -> A e. Fin ) )
8	6, 7	ax-mp 5	. . . . 5 |- ( A = (/) -> A e. Fin )
9	8	ancli 574	. . . 4 |- ( A = (/) -> ( A = (/) /\ A e. Fin ) )
10	5, 9	impbii 199	. . 3 |- ( ( A = (/) /\ A e. Fin ) <-> A = (/) )
11		rp-isfinite5 37863	. . . . . 6 |- ( A e. Fin <-> E. n e. NN0 ( 1 ... n ) ~~ A )
12		df-rex 2918	. . . . . 6 |- ( E. n e. NN0 ( 1 ... n ) ~~ A <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
13	11, 12	bitri 264	. . . . 5 |- ( A e. Fin <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
14	13	anbi2i 730	. . . 4 |- ( ( -. A = (/) /\ A e. Fin ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
15		df-rex 2918	. . . . 5 |- ( E. n e. NN ( 1 ... n ) ~~ A <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
16		en0 8019	. . . . . . . . . . . . . . 15 |- ( A ~~ (/) <-> A = (/) )
17	16	bicomi 214	. . . . . . . . . . . . . 14 |- ( A = (/) <-> A ~~ (/) )
18		ensymb 8004	. . . . . . . . . . . . . 14 |- ( A ~~ (/) <-> (/) ~~ A )
19	17, 18	bitri 264	. . . . . . . . . . . . 13 |- ( A = (/) <-> (/) ~~ A )
20	19	notbii 310	. . . . . . . . . . . 12 |- ( -. A = (/) <-> -. (/) ~~ A )
21		elnn0 11294	. . . . . . . . . . . . . 14 |- ( n e. NN0 <-> ( n e. NN \/ n = 0 ) )
22	21	anbi1i 731	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) )
23		andir 912	. . . . . . . . . . . . 13 |- ( ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
24	22, 23	bitri 264	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
25	20, 24	anbi12i 733	. . . . . . . . . . 11 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. (/) ~~ A /\ ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
26		andi 911	. . . . . . . . . . 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 264	. . . . . . . . . 10 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
28		3anass 1042	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) )
29		3anass 1042	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
30	28, 29	orbi12i 543	. . . . . . . . . 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 ) ) ) )
31	27, 30	sylbb2 228	. . . . . . . . 9 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) )
32		simp2 1062	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) -> n e. NN )
33		oveq2 6658	. . . . . . . . . . . 12 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
34		fz10 12362	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
35	33, 34	syl6eq 2672	. . . . . . . . . . 11 |- ( n = 0 -> ( 1 ... n ) = (/) )
36		simp2 1062	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) = (/) )
37		simp3 1063	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) ~~ A )
38	36, 37	eqbrtrrd 4677	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> (/) ~~ A )
39		simp1 1061	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
40	38, 39	pm2.21dd 186	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> n e. NN )
41	35, 40	syl3an2 1360	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) -> n e. NN )
42	32, 41	jaoi 394	. . . . . . . . 9 |- ( ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
43	31, 42	syl 17	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
44		simprr 796	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( 1 ... n ) ~~ A )
45	43, 44	jca 554	. . . . . . 7 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
46		nngt0 11049	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
47		hash0 13158	. . . . . . . . . . . . 13 |- ( # ` (/) ) = 0
48	47	a1i 11	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` (/) ) = 0 )
49		nnnn0 11299	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
50		hashfz1 13134	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( # ` ( 1 ... n ) ) = n )
51	49, 50	syl 17	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` ( 1 ... n ) ) = n )
52	46, 48, 51	3brtr4d 4685	. . . . . . . . . . 11 |- ( n e. NN -> ( # ` (/) ) < ( # ` ( 1 ... n ) ) )
53		fzfi 12771	. . . . . . . . . . . 12 |- ( 1 ... n ) e. Fin
54		hashsdom 13170	. . . . . . . . . . . 12 |- ( ( (/) e. Fin /\ ( 1 ... n ) e. Fin ) -> ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) ) )
55	6, 53, 54	mp2an 708	. . . . . . . . . . 11 |- ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) )
56	52, 55	sylib 208	. . . . . . . . . 10 |- ( n e. NN -> (/) ~< ( 1 ... n ) )
57	56	anim1i 592	. . . . . . . . 9 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) )
58		sdomentr 8094	. . . . . . . . . . 11 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> (/) ~< A )
59		sdomnen 7984	. . . . . . . . . . 11 |- ( (/) ~< A -> -. (/) ~~ A )
60	58, 59	syl 17	. . . . . . . . . 10 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
61		ensymb 8004	. . . . . . . . . . . 12 |- ( (/) ~~ A <-> A ~~ (/) )
62	61, 16	bitri 264	. . . . . . . . . . 11 |- ( (/) ~~ A <-> A = (/) )
63	62	notbii 310	. . . . . . . . . 10 |- ( -. (/) ~~ A <-> -. A = (/) )
64	60, 63	sylib 208	. . . . . . . . 9 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
65	57, 64	syl 17	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
66	49	anim1i 592	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
67	65, 66	jca 554	. . . . . . 7 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
68	45, 67	impbii 199	. . . . . 6 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
69	68	exbii 1774	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
70		19.42v 1918	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
71	15, 69, 70	3bitr2ri 289	. . . 4 |- ( ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n e. NN ( 1 ... n ) ~~ A )
72	14, 71	bitri 264	. . 3 |- ( ( -. A = (/) /\ A e. Fin ) <-> E. n e. NN ( 1 ... n ) ~~ A )
73	10, 72	orbi12i 543	. 2 |- ( ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )
74	4, 73	bitri 264	1 |- ( A e. Fin <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E.wrex 2913   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   ~~ cen 7952   ~< csdm 7954   Fincfn 7955   0cc0 9936   1c1 9937   < clt 10074   NNcn 11020   NN0cn0 11292   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118

This theorem is referenced by: (None)