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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
StepHypRef Expression
1 exmid 416 . . . 4  |-  ( A  =  (/)  \/  -.  A  =  (/) )
21biantrur 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 ) ) )
42, 3bitri 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 ) )
86, 7ax-mp 5 . . . . 5  |-  ( A  =  (/)  ->  A  e. 
Fin )
98ancli 553 . . . 4  |-  ( A  =  (/)  ->  ( A  =  (/)  /\  A  e. 
Fin ) )
105, 9impbii 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 ) )
1311, 12bitri 252 . . . . 5  |-  ( A  e.  Fin  <->  E. n
( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)
1413anbi2i 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  =  (/) )
1716bicomi 205 . . . . . . . . . . . . . 14  |-  ( A  =  (/)  <->  A  ~~  (/) )
18 ensymb 7571 . . . . . . . . . . . . . 14  |-  ( A 
~~  (/)  <->  (/)  ~~  A )
1917, 18bitri 252 . . . . . . . . . . . . 13  |-  ( A  =  (/)  <->  (/)  ~~  A )
2019notbii 297 . . . . . . . . . . . 12  |-  ( -.  A  =  (/)  <->  -.  (/)  ~~  A
)
21 elnn0 10822 . . . . . . . . . . . . . 14  |-  ( n  e.  NN0  <->  ( n  e.  NN  \/  n  =  0 ) )
2221anbi1i 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
) ) )
2422, 23bitri 252 . . . . . . . . . . . 12  |-  ( ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )  <->  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  \/  ( n  =  0  /\  ( 1 ... n )  ~~  A
) ) )
2520, 24anbi12i 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 )
) ) )
2725, 26bitri 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 )
) )
3028, 29orbi12i 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 )
) ) )
3130bicomi 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 )
) )
3227, 31sylbb 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 )  =  (/)
3634, 35syl6eq 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 )
3937, 38eqbrtrrd 4389 . . . . . . . . . . . 12  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  (/)  ~~  A
)
40 simp1 1005 . . . . . . . . . . . 12  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  -.  (/)  ~~  A )
4139, 40pm2.21dd 177 . . . . . . . . . . 11  |-  ( ( -.  (/)  ~~  A  /\  ( 1 ... n
)  =  (/)  /\  (
1 ... n )  ~~  A )  ->  n  e.  NN )
4236, 41syl3an2 1298 . . . . . . . . . 10  |-  ( ( -.  (/)  ~~  A  /\  n  =  0  /\  ( 1 ... n
)  ~~  A )  ->  n  e.  NN )
4333, 42jaoi 380 . . . . . . . . 9  |-  ( ( ( -.  (/)  ~~  A  /\  n  e.  NN  /\  ( 1 ... n
)  ~~  A )  \/  ( -.  (/)  ~~  A  /\  n  =  0  /\  ( 1 ... n
)  ~~  A )
)  ->  n  e.  NN )
4432, 43syl 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 )
4644, 45jca 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
4948a1i 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 )
5250, 51syl 17 . . . . . . . . . . . 12  |-  ( n  e.  NN  ->  ( # `
 ( 1 ... n ) )  =  n )
5347, 49, 523brtr4d 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
) ) )
566, 54, 55mp2an 676 . . . . . . . . . . 11  |-  ( (
# `  (/) )  < 
( # `  ( 1 ... n ) )  <->  (/) 
~<  ( 1 ... n
) )
5753, 56sylib 199 . . . . . . . . . 10  |-  ( n  e.  NN  ->  (/)  ~<  (
1 ... n ) )
5857anim1i 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
)
6159, 60syl 17 . . . . . . . . . 10  |-  ( (
(/)  ~<  ( 1 ... n )  /\  (
1 ... n )  ~~  A )  ->  -.  (/)  ~~  A )
62 ensymb 7571 . . . . . . . . . . . 12  |-  ( (/)  ~~  A  <->  A  ~~  (/) )
6362, 16bitri 252 . . . . . . . . . . 11  |-  ( (/)  ~~  A  <->  A  =  (/) )
6463notbii 297 . . . . . . . . . 10  |-  ( -.  (/)  ~~  A  <->  -.  A  =  (/) )
6561, 64sylib 199 . . . . . . . . 9  |-  ( (
(/)  ~<  ( 1 ... n )  /\  (
1 ... n )  ~~  A )  ->  -.  A  =  (/) )
6658, 65syl 17 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  -.  A  =  (/) )
6750anim1i 570 . . . . . . . 8  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)
6866, 67jca 534 . . . . . . 7  |-  ( ( n  e.  NN  /\  ( 1 ... n
)  ~~  A )  ->  ( -.  A  =  (/)  /\  ( n  e. 
NN0  /\  ( 1 ... n )  ~~  A ) ) )
6946, 68impbii 190 . . . . . 6  |-  ( ( -.  A  =  (/)  /\  ( n  e.  NN0  /\  ( 1 ... n
)  ~~  A )
)  <->  ( n  e.  NN  /\  ( 1 ... n )  ~~  A ) )
7069exbii 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 ) ) )
7215, 70, 713bitr2ri 277 . . . 4  |-  ( ( -.  A  =  (/)  /\ 
E. n ( n  e.  NN0  /\  (
1 ... n )  ~~  A ) )  <->  E. n  e.  NN  ( 1 ... n )  ~~  A
)
7314, 72bitri 252 . . 3  |-  ( ( -.  A  =  (/)  /\  A  e.  Fin )  <->  E. n  e.  NN  (
1 ... n )  ~~  A )
7410, 73orbi12i 523 . 2  |-  ( ( ( A  =  (/)  /\  A  e.  Fin )  \/  ( -.  A  =  (/)  /\  A  e.  Fin ) )  <->  ( A  =  (/)  \/  E. n  e.  NN  ( 1 ... n )  ~~  A
) )
754, 74bitri 252 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 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)
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