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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
StepHypRef Expression
1 exmid 413 . . . 4  |-  ( A  =  (/)  \/  -.  A  =  (/) )
21biantrur 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 ) ) )
42, 3bitri 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 ) )
86, 7ax-mp 5 . . . . 5  |-  ( A  =  (/)  ->  A  e. 
Fin )
98ancli 549 . . . 4  |-  ( A  =  (/)  ->  ( A  =  (/)  /\  A  e. 
Fin ) )
105, 9impbii 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 ) )
1311, 12bitri 249 . . . . 5  |-  ( A  e.  Fin  <->  E. n
( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)
1413anbi2i 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  =  (/) )
1716bicomi 202 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  <->  A  ~~  (/) )
18 ensymb 7556 . . . . . . . . . . . . . 14  |-  ( A 
~~  (/)  <->  (/)  ~~  A )
1917, 18bitri 249 . . . . . . . . . . . . 13  |-  ( A  =  (/)  <->  (/)  ~~  A )
2019notbii 294 . . . . . . . . . . . 12  |-  ( -.  A  =  (/)  <->  -.  (/)  ~~  A
)
21 elnn0 10793 . . . . . . . . . . . . . 14  |-  ( n  e.  NN0  <->  ( n  e.  NN  \/  n  =  0 ) )
2221anbi1i 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
) ) )
2422, 23bitri 249 . . . . . . . . . . . 12  |-  ( ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )  <->  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  \/  ( n  =  0  /\  ( 1 ... n )  ~~  A
) ) )
2520, 24anbi12i 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 )
) ) )
2725, 26bitri 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 )
) )
3028, 29orbi12i 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 )
) ) )
3130bicomi 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 )
) )
3227, 31sylbb 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 )  =  (/)
3634, 35syl6eq 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 )
3937, 38eqbrtrrd 4461 . . . . . . . . . . . 12  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  (/)  ~~  A
)
40 simp1 994 . . . . . . . . . . . 12  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  -.  (/)  ~~  A )
4139, 40pm2.21dd 174 . . . . . . . . . . 11  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  n  e.  NN )
4236, 41syl3an2 1260 . . . . . . . . . 10  |-  ( ( -.  (/)  ~~  A  /\  n  =  0  /\  ( 1 ... n
)  ~~  A )  ->  n  e.  NN )
4333, 42jaoi 377 . . . . . . . . 9  |-  ( ( ( -.  (/)  ~~  A  /\  n  e.  NN  /\  ( 1 ... n
)  ~~  A )  \/  ( -.  (/)  ~~  A  /\  n  =  0  /\  ( 1 ... n
)  ~~  A )
)  ->  n  e.  NN )
4432, 43syl 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 )
4644, 45jca 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
4948a1i 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 )
5250, 51syl 16 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
5347, 49, 523brtr4d 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
) ) )
566, 54, 55mp2an 670 . . . . . . . . . . 11  |-  ( (
# `  (/) )  < 
( # `  ( 1 ... n ) )  <->  (/) 
~<  ( 1 ... n
) )
5753, 56sylib 196 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (/)  ~<  (
1 ... n ) )
5857anim1i 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
)
6159, 60syl 16 . . . . . . . . . 10  |-  ( (
(/)  ~<  ( 1 ... n )  /\  (
1 ... n )  ~~  A )  ->  -.  (/)  ~~  A )
62 ensymb 7556 . . . . . . . . . . . 12  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6362, 16bitri 249 . . . . . . . . . . 11  |-  ( (/)  ~~  A  <->  A  =  (/) )
6463notbii 294 . . . . . . . . . 10  |-  ( -.  (/)  ~~  A  <->  -.  A  =  (/) )
6561, 64sylib 196 . . . . . . . . 9  |-  ( (
(/)  ~<  ( 1 ... n )  /\  (
1 ... n )  ~~  A )  ->  -.  A  =  (/) )
6658, 65syl 16 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  -.  A  =  (/) )
6750anim1i 566 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)
6866, 67jca 530 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  ( -.  A  =  (/)  /\  ( n  e. 
NN0  /\  ( 1 ... n )  ~~  A ) ) )
6946, 68impbii 188 . . . . . 6  |-  ( ( -.  A  =  (/)  /\  ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)  <->  ( n  e.  NN  /\  ( 1 ... n )  ~~  A ) )
7069exbii 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 ) ) )
7215, 70, 713bitr2ri 274 . . . 4  |-  ( ( -.  A  =  (/)  /\ 
E. n ( n  e.  NN0  /\  (
1 ... n )  ~~  A ) )  <->  E. n  e.  NN  ( 1 ... n )  ~~  A
)
7314, 72bitri 249 . . 3  |-  ( ( -.  A  =  (/)  /\  A  e.  Fin )  <->  E. n  e.  NN  (
1 ... n )  ~~  A )
7410, 73orbi12i 519 . 2  |-  ( ( ( A  =  (/)  /\  A  e.  Fin )  \/  ( -.  A  =  (/)  /\  A  e.  Fin ) )  <->  ( A  =  (/)  \/  E. n  e.  NN  ( 1 ... n )  ~~  A
) )
754, 74bitri 249 1  |-  ( A  e.  Fin  <->  ( A  =  (/)  \/  E. n  e.  NN  ( 1 ... n )  ~~  A
) )
Colors of variables: wff setvar class
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)
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