Theorem rp-isfinite6 38157

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 413	. . . 4 |- ( A = (/) \/ -. A = (/) )
2	1	biantrur 504	. . 3 |- ( A e. Fin <-> ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) )
3		andir 866	. . 3 |- ( ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
4	2, 3	bitri 249	. 2 |- ( A e. Fin <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
5		simpl 455	. . . 4 |- ( ( A = (/) /\ A e. Fin ) -> A = (/) )
6		0fin 7740	. . . . . 6 |- (/) e. Fin
7		eleq1a 2537	. . . . . 6 |- ( (/) e. Fin -> ( A = (/) -> A e. Fin ) )
8	6, 7	ax-mp 5	. . . . 5 |- ( A = (/) -> A e. Fin )
9	8	ancli 549	. . . 4 |- ( A = (/) -> ( A = (/) /\ A e. Fin ) )
10	5, 9	impbii 188	. . 3 |- ( ( A = (/) /\ A e. Fin ) <-> A = (/) )
11		rp-isfinite5 38156	. . . . . 6 |- ( A e. Fin <-> E. n e. NN0 ( 1 ... n ) ~~ A )
12		df-rex 2810	. . . . . 6 |- ( E. n e. NN0 ( 1 ... n ) ~~ A <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
13	11, 12	bitri 249	. . . . 5 |- ( A e. Fin <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
14	13	anbi2i 692	. . . 4 |- ( ( -. A = (/) /\ A e. Fin ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
15		df-rex 2810	. . . . 5 |- ( E. n e. NN ( 1 ... n ) ~~ A <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
16		en0 7571	. . . . . . . . . . . . . . 15 |- ( A ~~ (/) <-> A = (/) )
17	16	bicomi 202	. . . . . . . . . . . . . 14 |- ( A = (/) <-> A ~~ (/) )
18		ensymb 7556	. . . . . . . . . . . . . 14 |- ( A ~~ (/) <-> (/) ~~ A )
19	17, 18	bitri 249	. . . . . . . . . . . . 13 |- ( A = (/) <-> (/) ~~ A )
20	19	notbii 294	. . . . . . . . . . . 12 |- ( -. A = (/) <-> -. (/) ~~ A )
21		elnn0 10793	. . . . . . . . . . . . . 14 |- ( n e. NN0 <-> ( n e. NN \/ n = 0 ) )
22	21	anbi1i 693	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) )
23		andir 866	. . . . . . . . . . . . 13 |- ( ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
24	22, 23	bitri 249	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
25	20, 24	anbi12i 695	. . . . . . . . . . 11 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. (/) ~~ A /\ ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
26		andi 865	. . . . . . . . . . 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 249	. . . . . . . . . 10 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
28		3anass 975	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) )
29		3anass 975	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
30	28, 29	orbi12i 519	. . . . . . . . . . 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 202	. . . . . . . . . 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 197	. . . . . . . . 9 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) )
33		simp2 995	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) -> n e. NN )
34		oveq2 6278	. . . . . . . . . . . 12 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
35		fz10 11709	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
36	34, 35	syl6eq 2511	. . . . . . . . . . 11 |- ( n = 0 -> ( 1 ... n ) = (/) )
37		simp2 995	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) = (/) )
38		simp3 996	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) ~~ A )
39	37, 38	eqbrtrrd 4461	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> (/) ~~ A )
40		simp1 994	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
41	39, 40	pm2.21dd 174	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> n e. NN )
42	36, 41	syl3an2 1260	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) -> n e. NN )
43	33, 42	jaoi 377	. . . . . . . . 9 |- ( ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
44	32, 43	syl 16	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> n e. NN )
45		simprr 755	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( 1 ... n ) ~~ A )
46	44, 45	jca 530	. . . . . . 7 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
47		nngt0 10560	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
48		hash0 12420	. . . . . . . . . . . . 13 |- ( # ` (/) ) = 0
49	48	a1i 11	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` (/) ) = 0 )
50		nnnn0 10798	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
51		hashfz1 12401	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( # ` ( 1 ... n ) ) = n )
52	50, 51	syl 16	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` ( 1 ... n ) ) = n )
53	47, 49, 52	3brtr4d 4469	. . . . . . . . . . 11 |- ( n e. NN -> ( # ` (/) ) < ( # ` ( 1 ... n ) ) )
54		fzfi 12064	. . . . . . . . . . . 12 |- ( 1 ... n ) e. Fin
55		hashsdom 12432	. . . . . . . . . . . 12 |- ( ( (/) e. Fin /\ ( 1 ... n ) e. Fin ) -> ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) ) )
56	6, 54, 55	mp2an 670	. . . . . . . . . . 11 |- ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) )
57	53, 56	sylib 196	. . . . . . . . . 10 |- ( n e. NN -> (/) ~< ( 1 ... n ) )
58	57	anim1i 566	. . . . . . . . 9 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) )
59		sdomentr 7644	. . . . . . . . . . 11 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> (/) ~< A )
60		sdomnen 7537	. . . . . . . . . . 11 |- ( (/) ~< A -> -. (/) ~~ A )
61	59, 60	syl 16	. . . . . . . . . 10 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
62		ensymb 7556	. . . . . . . . . . . 12 |- ( (/) ~~ A <-> A ~~ (/) )
63	62, 16	bitri 249	. . . . . . . . . . 11 |- ( (/) ~~ A <-> A = (/) )
64	63	notbii 294	. . . . . . . . . 10 |- ( -. (/) ~~ A <-> -. A = (/) )
65	61, 64	sylib 196	. . . . . . . . 9 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
66	58, 65	syl 16	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
67	50	anim1i 566	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
68	66, 67	jca 530	. . . . . . 7 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
69	46, 68	impbii 188	. . . . . 6 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
70	69	exbii 1672	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
71		19.42v 1780	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
72	15, 70, 71	3bitr2ri 274	. . . 4 |- ( ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n e. NN ( 1 ... n ) ~~ A )
73	14, 72	bitri 249	. . 3 |- ( ( -. A = (/) /\ A e. Fin ) <-> E. n e. NN ( 1 ... n ) ~~ A )
74	10, 73	orbi12i 519	. 2 |- ( ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )
75	4, 74	bitri 249	1 |- ( A e. Fin <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 366   /\ wa 367   /\ w3a 971   = wceq 1398   E.wex 1617   e. wcel 1823   E.wrex 2805   (/)c0 3783   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   ~~ cen 7506   ~< csdm 7508   Fincfn 7509   0cc0 9481   1c1 9482   < clt 9617   NNcn 10531   NN0cn0 10791   ...cfz 11675   #chash 12387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388

This theorem is referenced by: (None)