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Theorem ennnfonelemr 11925
Description: Lemma for ennnfone 11927. 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 1688 . . . . 5  |-  ( x  =  a  ->  (
x  =  y  <->  a  =  y ) )
32dcbid 823 . . . 4  |-  ( x  =  a  ->  (DECID  x  =  y  <-> DECID  a  =  y )
)
4 equequ2 1689 . . . . 5  |-  ( y  =  b  ->  (
a  =  y  <->  a  =  b ) )
54dcbid 823 . . . 4  |-  ( y  =  b  ->  (DECID  a  =  y  <-> DECID  a  =  b )
)
63, 5cbvral2v 2660 . . 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 121 . 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 5414 . . . . . . . . 9  |-  ( j  =  f  ->  ( F `  j )  =  ( F `  f ) )
1110neeq2d 2325 . . . . . . . 8  |-  ( j  =  f  ->  (
( F `  k
)  =/=  ( F `
 j )  <->  ( F `  k )  =/=  ( F `  f )
) )
1211cbvralv 2652 . . . . . . 7  |-  ( A. j  e.  ( 0 ... n ) ( F `  k )  =/=  ( F `  j )  <->  A. f  e.  ( 0 ... n
) ( F `  k )  =/=  ( F `  f )
)
1312rexbii 2440 . . . . . 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 5414 . . . . . . . . 9  |-  ( k  =  e  ->  ( F `  k )  =  ( F `  e ) )
1514neeq1d 2324 . . . . . . . 8  |-  ( k  =  e  ->  (
( F `  k
)  =/=  ( F `
 f )  <->  ( F `  e )  =/=  ( F `  f )
) )
1615ralbidv 2435 . . . . . . 7  |-  ( k  =  e  ->  ( A. f  e.  (
0 ... n ) ( F `  k )  =/=  ( F `  f )  <->  A. f  e.  ( 0 ... n
) ( F `  e )  =/=  ( F `  f )
) )
1716cbvrexv 2653 . . . . . 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 183 . . . . 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 2439 . . . 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 5775 . . . . . . 7  |-  ( n  =  d  ->  (
0 ... n )  =  ( 0 ... d
) )
2120raleqdv 2630 . . . . . 6  |-  ( n  =  d  ->  ( A. f  e.  (
0 ... n ) ( F `  e )  =/=  ( F `  f )  <->  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
) )
2221rexbidv 2436 . . . . 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 2652 . . . 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 183 . . 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 121 . 2  |-  ( ph  ->  A. d  e.  NN0  E. e  e.  NN0  A. f  e.  ( 0 ... d
) ( F `  e )  =/=  ( F `  f )
)
26 oveq1 5774 . . . 4  |-  ( c  =  a  ->  (
c  +  1 )  =  ( a  +  1 ) )
2726cbvmptv 4019 . . 3  |-  ( c  e.  ZZ  |->  ( c  +  1 ) )  =  ( a  e.  ZZ  |->  ( a  +  1 ) )
28 freceq1 6282 . . 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 11924 1  |-  ( ph  ->  A  ~~  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4  DECID wdc 819    = wceq 1331    =/= wne 2306   A.wral 2414   E.wrex 2415   class class class wbr 3924    |-> cmpt 3984   -onto->wfo 5116   ` cfv 5118  (class class class)co 5767  freccfrec 6280    ~~ cen 6625   0cc0 7613   1c1 7614    + caddc 7616   NNcn 8713   NN0cn0 8970   ZZcz 9047   ...cfz 9783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-ltadd 7729
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-er 6422  df-pm 6538  df-en 6628  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-seqfrec 10212
This theorem is referenced by:  ennnfone  11927
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