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Theorem ennnfone 12426
Description: A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that 
A is countable (that's the  f : NN0 -onto-> A part, as seen in theorems like ctm 7108), infinite (that's the part about being able to find an element of  A distinct from any mapping of a natural number via  f), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
Assertion
Ref Expression
ennnfone  |-  ( A 
~~  NN  <->  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. f ( f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) ) )
Distinct variable groups:    A, f, j, n, x, y    f,
k, j, n
Allowed substitution hint:    A( k)

Proof of Theorem ennnfone
StepHypRef Expression
1 ennnfonelemim 12425 . 2  |-  ( A 
~~  NN  ->  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. f ( f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) ) )
2 simpl 109 . . . . . 6  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  (
f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) )  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
3 simprl 529 . . . . . 6  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  (
f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) )  ->  f : NN0 -onto-> A )
4 simprr 531 . . . . . 6  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  (
f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) )  ->  A. n  e.  NN0  E. k  e. 
NN0  A. j  e.  ( 0 ... n ) ( f `  k
)  =/=  ( f `
 j ) )
52, 3, 4ennnfonelemr 12424 . . . . 5  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  (
f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) )  ->  A  ~~  NN )
65ex 115 . . . 4  |-  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  ->  ( ( f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
)  ->  A  ~~  NN ) )
76exlimdv 1819 . . 3  |-  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  ->  ( E. f
( f : NN0 -onto-> A  /\  A. n  e. 
NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n ) ( f `  k )  =/=  ( f `  j ) )  ->  A  ~~  NN ) )
87imp 124 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. f
( f : NN0 -onto-> A  /\  A. n  e. 
NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n ) ( f `  k )  =/=  ( f `  j ) ) )  ->  A  ~~  NN )
91, 8impbii 126 1  |-  ( A 
~~  NN  <->  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  E. f ( f : NN0 -onto-> A  /\  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( f `  k )  =/=  (
f `  j )
) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105  DECID wdc 834   E.wex 1492    =/= wne 2347   A.wral 2455   E.wrex 2456   class class class wbr 4004   -onto->wfo 5215   ` cfv 5217  (class class class)co 5875    ~~ cen 6738   0cc0 7811   NNcn 8919   NN0cn0 9176   ...cfz 10008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-er 6535  df-pm 6651  df-en 6741  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-fz 10009  df-seqfrec 10446
This theorem is referenced by:  ctinfom  12429
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