Theorem rp-isfinite6 36208

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 421	. . . 4 |- ( A = (/) \/ -. A = (/) )
2	1	biantrur 513	. . 3 |- ( A e. Fin <-> ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) )
3		andir 884	. . 3 |- ( ( ( A = (/) \/ -. A = (/) ) /\ A e. Fin ) <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
4	2, 3	bitri 257	. 2 |- ( A e. Fin <-> ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) )
5		simpl 463	. . . 4 |- ( ( A = (/) /\ A e. Fin ) -> A = (/) )
6		0fin 7825	. . . . . 6 |- (/) e. Fin
7		eleq1a 2535	. . . . . 6 |- ( (/) e. Fin -> ( A = (/) -> A e. Fin ) )
8	6, 7	ax-mp 5	. . . . 5 |- ( A = (/) -> A e. Fin )
9	8	ancli 558	. . . 4 |- ( A = (/) -> ( A = (/) /\ A e. Fin ) )
10	5, 9	impbii 192	. . 3 |- ( ( A = (/) /\ A e. Fin ) <-> A = (/) )
11		rp-isfinite5 36207	. . . . . 6 |- ( A e. Fin <-> E. n e. NN0 ( 1 ... n ) ~~ A )
12		df-rex 2755	. . . . . 6 |- ( E. n e. NN0 ( 1 ... n ) ~~ A <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
13	11, 12	bitri 257	. . . . 5 |- ( A e. Fin <-> E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
14	13	anbi2i 705	. . . 4 |- ( ( -. A = (/) /\ A e. Fin ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
15		df-rex 2755	. . . . 5 |- ( E. n e. NN ( 1 ... n ) ~~ A <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
16		en0 7658	. . . . . . . . . . . . . . 15 |- ( A ~~ (/) <-> A = (/) )
17	16	bicomi 207	. . . . . . . . . . . . . 14 |- ( A = (/) <-> A ~~ (/) )
18		ensymb 7643	. . . . . . . . . . . . . 14 |- ( A ~~ (/) <-> (/) ~~ A )
19	17, 18	bitri 257	. . . . . . . . . . . . 13 |- ( A = (/) <-> (/) ~~ A )
20	19	notbii 302	. . . . . . . . . . . 12 |- ( -. A = (/) <-> -. (/) ~~ A )
21		elnn0 10900	. . . . . . . . . . . . . 14 |- ( n e. NN0 <-> ( n e. NN \/ n = 0 ) )
22	21	anbi1i 706	. . . . . . . . . . . . 13 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) )
23		andir 884	. . . . . . . . . . . . 13 |- ( ( ( n e. NN \/ n = 0 ) /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
24	22, 23	bitri 257	. . . . . . . . . . . 12 |- ( ( n e. NN0 /\ ( 1 ... n ) ~~ A ) <-> ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
25	20, 24	anbi12i 708	. . . . . . . . . . 11 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. (/) ~~ A /\ ( ( n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
26		andi 883	. . . . . . . . . . 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 257	. . . . . . . . . 10 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) \/ ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) ) )
28		3anass 995	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n e. NN /\ ( 1 ... n ) ~~ A ) ) )
29		3anass 995	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) <-> ( -. (/) ~~ A /\ ( n = 0 /\ ( 1 ... n ) ~~ A ) ) )
30	28, 29	orbi12i 528	. . . . . . . . . . 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 207	. . . . . . . . . 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 202	. . . . . . . . 9 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) \/ ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) ) )
33		simp2 1015	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n e. NN /\ ( 1 ... n ) ~~ A ) -> n e. NN )
34		oveq2 6323	. . . . . . . . . . . 12 |- ( n = 0 -> ( 1 ... n ) = ( 1 ... 0 ) )
35		fz10 11849	. . . . . . . . . . . 12 |- ( 1 ... 0 ) = (/)
36	34, 35	syl6eq 2512	. . . . . . . . . . 11 |- ( n = 0 -> ( 1 ... n ) = (/) )
37		simp2 1015	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) = (/) )
38		simp3 1016	. . . . . . . . . . . . 13 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> ( 1 ... n ) ~~ A )
39	37, 38	eqbrtrrd 4439	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> (/) ~~ A )
40		simp1 1014	. . . . . . . . . . . 12 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
41	39, 40	pm2.21dd 179	. . . . . . . . . . 11 |- ( ( -. (/) ~~ A /\ ( 1 ... n ) = (/) /\ ( 1 ... n ) ~~ A ) -> n e. NN )
42	36, 41	syl3an2 1310	. . . . . . . . . 10 |- ( ( -. (/) ~~ A /\ n = 0 /\ ( 1 ... n ) ~~ A ) -> n e. NN )
43	33, 42	jaoi 385	. . . . . . . . 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 771	. . . . . . . 8 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( 1 ... n ) ~~ A )
46	44, 45	jca 539	. . . . . . 7 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) -> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
47		nngt0 10666	. . . . . . . . . . . 12 |- ( n e. NN -> 0 < n )
48		hash0 12580	. . . . . . . . . . . . 13 |- ( # ` (/) ) = 0
49	48	a1i 11	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` (/) ) = 0 )
50		nnnn0 10905	. . . . . . . . . . . . 13 |- ( n e. NN -> n e. NN0 )
51		hashfz1 12561	. . . . . . . . . . . . 13 |- ( n e. NN0 -> ( # ` ( 1 ... n ) ) = n )
52	50, 51	syl 17	. . . . . . . . . . . 12 |- ( n e. NN -> ( # ` ( 1 ... n ) ) = n )
53	47, 49, 52	3brtr4d 4447	. . . . . . . . . . 11 |- ( n e. NN -> ( # ` (/) ) < ( # ` ( 1 ... n ) ) )
54		fzfi 12217	. . . . . . . . . . . 12 |- ( 1 ... n ) e. Fin
55		hashsdom 12592	. . . . . . . . . . . 12 |- ( ( (/) e. Fin /\ ( 1 ... n ) e. Fin ) -> ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) ) )
56	6, 54, 55	mp2an 683	. . . . . . . . . . 11 |- ( ( # ` (/) ) < ( # ` ( 1 ... n ) ) <-> (/) ~< ( 1 ... n ) )
57	53, 56	sylib 201	. . . . . . . . . 10 |- ( n e. NN -> (/) ~< ( 1 ... n ) )
58	57	anim1i 576	. . . . . . . . 9 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) )
59		sdomentr 7732	. . . . . . . . . . 11 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> (/) ~< A )
60		sdomnen 7624	. . . . . . . . . . 11 |- ( (/) ~< A -> -. (/) ~~ A )
61	59, 60	syl 17	. . . . . . . . . 10 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. (/) ~~ A )
62		ensymb 7643	. . . . . . . . . . . 12 |- ( (/) ~~ A <-> A ~~ (/) )
63	62, 16	bitri 257	. . . . . . . . . . 11 |- ( (/) ~~ A <-> A = (/) )
64	63	notbii 302	. . . . . . . . . 10 |- ( -. (/) ~~ A <-> -. A = (/) )
65	61, 64	sylib 201	. . . . . . . . 9 |- ( ( (/) ~< ( 1 ... n ) /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
66	58, 65	syl 17	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> -. A = (/) )
67	50	anim1i 576	. . . . . . . 8 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( n e. NN0 /\ ( 1 ... n ) ~~ A ) )
68	66, 67	jca 539	. . . . . . 7 |- ( ( n e. NN /\ ( 1 ... n ) ~~ A ) -> ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
69	46, 68	impbii 192	. . . . . 6 |- ( ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( n e. NN /\ ( 1 ... n ) ~~ A ) )
70	69	exbii 1729	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n ( n e. NN /\ ( 1 ... n ) ~~ A ) )
71		19.42v 1845	. . . . 5 |- ( E. n ( -. A = (/) /\ ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) )
72	15, 70, 71	3bitr2ri 282	. . . 4 |- ( ( -. A = (/) /\ E. n ( n e. NN0 /\ ( 1 ... n ) ~~ A ) ) <-> E. n e. NN ( 1 ... n ) ~~ A )
73	14, 72	bitri 257	. . 3 |- ( ( -. A = (/) /\ A e. Fin ) <-> E. n e. NN ( 1 ... n ) ~~ A )
74	10, 73	orbi12i 528	. 2 |- ( ( ( A = (/) /\ A e. Fin ) \/ ( -. A = (/) /\ A e. Fin ) ) <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )
75	4, 74	bitri 257	1 |- ( A e. Fin <-> ( A = (/) \/ E. n e. NN ( 1 ... n ) ~~ A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   /\ w3a 991   = wceq 1455   E.wex 1674   e. wcel 1898   E.wrex 2750   (/)c0 3743   class class class wbr 4416   ` cfv 5601  (class class class)co 6315   ~~ cen 7592   ~< csdm 7594   Fincfn 7595   0cc0 9565   1c1 9566   < clt 9701   NNcn 10637   NN0cn0 10898   ...cfz 11813   #chash 12547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-om 6720  df-1st 6820  df-2nd 6821  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-oadd 7212  df-er 7389  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-card 8399  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-nn 10638  df-n0 10899  df-z 10967  df-uz 11189  df-fz 11814  df-hash 12548

This theorem is referenced by: (None)