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Theorem ennnfonelemr 12449
Description: Lemma for ennnfone 12451. The interesting direction, expressed in deduction form. (Contributed by Jim Kingdon, 27-Oct-2022.)
Hypotheses
Ref Expression
ennnfonelemr.dceq  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
ennnfonelemr.f  |-  ( ph  ->  F : NN0 -onto-> A
)
ennnfonelemr.n  |-  ( ph  ->  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )
)
Assertion
Ref Expression
ennnfonelemr  |-  ( ph  ->  A  ~~  NN )
Distinct variable groups:    y, A, x   
n, F, j, k
Allowed substitution hints:    ph( x, y, j, k, n)    A( j,
k, n)    F( x, y)

Proof of Theorem ennnfonelemr
Dummy variables  a  b  d  e  f  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ennnfonelemr.dceq . . 3  |-  ( ph  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y )
2 equequ1 1723 . . . . 5  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
32dcbid 839 . . . 4  |-  ( x  =  a  ->  (DECID  x  =  y  <-> DECID  a  =  y )
)
4 equequ2 1724 . . . . 5  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
54dcbid 839 . . . 4  |-  ( y  =  b  ->  (DECID  a  =  y  <-> DECID  a  =  b )
)
63, 5cbvral2v 2731 . . 3  |-  ( A. x  e.  A  A. y  e.  A DECID  x  =  y 
<-> 
A. a  e.  A  A. b  e.  A DECID  a  =  b )
71, 6sylib 122 . 2  |-  ( ph  ->  A. a  e.  A  A. b  e.  A DECID  a  =  b )
8 ennnfonelemr.f . 2  |-  ( ph  ->  F : NN0 -onto-> A
)
9 ennnfonelemr.n . . 3  |-  ( ph  ->  A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )
)
10 fveq2 5531 . . . . . . . . 9  |-  ( j  =  f  ->  ( F `  j )  =  ( F `  f ) )
1110neeq2d 2379 . . . . . . . 8  |-  ( j  =  f  ->  (
( F `  k
)  =/=  ( F `
 j )  <->  ( F `  k )  =/=  ( F `  f )
) )
1211cbvralv 2718 . . . . . . 7  |-  ( A. j  e.  ( 0 ... n ) ( F `  k )  =/=  ( F `  j )  <->  A. f  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  f )
)
1312rexbii 2497 . . . . . 6  |-  ( E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )  <->  E. k  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  f )
)
14 fveq2 5531 . . . . . . . . 9  |-  ( k  =  e  ->  ( F `  k )  =  ( F `  e ) )
1514neeq1d 2378 . . . . . . . 8  |-  ( k  =  e  ->  (
( F `  k
)  =/=  ( F `
 f )  <->  ( F `  e )  =/=  ( F `  f )
) )
1615ralbidv 2490 . . . . . . 7  |-  ( k  =  e  ->  ( A. f  e.  (
0 ... n ) ( F `  k )  =/=  ( F `  f )  <->  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )
) )
1716cbvrexv 2719 . . . . . 6  |-  ( E. k  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  f )  <->  E. e  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )
)
1813, 17bitri 184 . . . . 5  |-  ( E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )  <->  E. e  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )
)
1918ralbii 2496 . . . 4  |-  ( A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )  <->  A. n  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )
)
20 oveq2 5900 . . . . . . 7  |-  ( n  =  d  ->  (
0 ... n )  =  ( 0 ... d
) )
2120raleqdv 2692 . . . . . 6  |-  ( n  =  d  ->  ( A. f  e.  (
0 ... n ) ( F `  e )  =/=  ( F `  f )  <->  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
) )
2221rexbidv 2491 . . . . 5  |-  ( n  =  d  ->  ( E. e  e.  NN0  A. f  e.  ( 0 ... n ) ( F `  e )  =/=  ( F `  f )  <->  E. e  e.  NN0  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
) )
2322cbvralv 2718 . . . 4  |-  ( A. n  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )  <->  A. d  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
)
2419, 23bitri 184 . . 3  |-  ( A. n  e.  NN0  E. k  e.  NN0  A. j  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  j )  <->  A. d  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
)
259, 24sylib 122 . 2  |-  ( ph  ->  A. d  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
)
26 oveq1 5899 . . . 4  |-  ( c  =  a  ->  (
c  +  1 )  =  ( a  +  1 ) )
2726cbvmptv 4114 . . 3  |-  ( c  e.  ZZ  |->  ( c  +  1 ) )  =  ( a  e.  ZZ  |->  ( a  +  1 ) )
28 freceq1 6412 . . 3  |-  ( ( c  e.  ZZ  |->  ( c  +  1 ) )  =  ( a  e.  ZZ  |->  ( a  +  1 ) )  -> frec ( ( c  e.  ZZ  |->  ( c  +  1 ) ) ,  0 )  = frec ( ( a  e.  ZZ  |->  ( a  +  1 ) ) ,  0 ) )
2927, 28ax-mp 5 . 2  |- frec ( ( c  e.  ZZ  |->  ( c  +  1 ) ) ,  0 )  = frec ( ( a  e.  ZZ  |->  ( a  +  1 ) ) ,  0 )
307, 8, 25, 29ennnfonelemnn0 12448 1  |-  ( ph  ->  A  ~~  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 835    = wceq 1364    =/= wne 2360   A.wral 2468   E.wrex 2469   class class class wbr 4018    |-> cmpt 4079   -onto->wfo 5230   ` cfv 5232  (class class class)co 5892  freccfrec 6410    ~~ cen 6757   0cc0 7831   1c1 7832    + caddc 7834   NNcn 8939   NN0cn0 9196   ZZcz 9273   ...cfz 10028
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7922  ax-resscn 7923  ax-1cn 7924  ax-1re 7925  ax-icn 7926  ax-addcl 7927  ax-addrcl 7928  ax-mulcl 7929  ax-addcom 7931  ax-addass 7933  ax-distr 7935  ax-i2m1 7936  ax-0lt1 7937  ax-0id 7939  ax-rnegex 7940  ax-cnre 7942  ax-pre-ltirr 7943  ax-pre-ltwlin 7944  ax-pre-lttrn 7945  ax-pre-ltadd 7947
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5234  df-fn 5235  df-f 5236  df-f1 5237  df-fo 5238  df-f1o 5239  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-1st 6160  df-2nd 6161  df-recs 6325  df-frec 6411  df-er 6554  df-pm 6670  df-en 6760  df-pnf 8014  df-mnf 8015  df-xr 8016  df-ltxr 8017  df-le 8018  df-sub 8150  df-neg 8151  df-inn 8940  df-n0 9197  df-z 9274  df-uz 9549  df-fz 10029  df-seqfrec 10466
This theorem is referenced by:  ennnfone  12451
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