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Theorem nn0disj 9922
Description: The first  N  + 
1 elements of the set of nonnegative integers are distinct from any later members. (Contributed by AV, 8-Nov-2019.)
Assertion
Ref Expression
nn0disj  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  =  (/)

Proof of Theorem nn0disj
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 elin 3259 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  <->  ( k  e.  ( 0 ... N
)  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) ) )
21simprbi 273 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
3 eluzle 9345 . . . . . 6  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  <_ 
k )
42, 3syl 14 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  <_ 
k )
5 eluzel2 9338 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  ( N  +  1 )  e.  ZZ )
62, 5syl 14 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  +  1 )  e.  ZZ )
7 eluzelz 9342 . . . . . . 7  |-  ( k  e.  ( ZZ>= `  ( N  +  1 ) )  ->  k  e.  ZZ )
82, 7syl 14 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ZZ )
9 zlem1lt 9117 . . . . . 6  |-  ( ( ( N  +  1 )  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( N  + 
1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  <  k ) )
106, 8, 9syl2anc 408 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  < 
k ) )
114, 10mpbid 146 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  < 
k )
121simplbi 272 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( 0 ... N
) )
13 elfzle2 9815 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  <_  N )
1412, 13syl 14 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  <_  N )
158zred 9180 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  RR )
16 elfzel2 9811 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716adantr 274 . . . . . . . . 9  |-  ( ( k  e.  ( 0 ... N )  /\  k  e.  ( ZZ>= `  ( N  +  1
) ) )  ->  N  e.  ZZ )
181, 17sylbi 120 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  ZZ )
1918zred 9180 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  RR )
2015, 19lenltd 7887 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  N  <  k ) )
2118zcnd 9181 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  CC )
22 pncan1 8146 . . . . . . . . . 10  |-  ( N  e.  CC  ->  (
( N  +  1 )  -  1 )  =  N )
2321, 22syl 14 . . . . . . . . 9  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( ( N  +  1 )  -  1 )  =  N )
2423eqcomd 2145 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  =  ( ( N  + 
1 )  -  1 ) )
2524breq1d 3939 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  <  k  <->  ( ( N  +  1 )  - 
1 )  <  k
) )
2625notbid 656 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( -.  N  <  k  <->  -.  (
( N  +  1 )  -  1 )  <  k ) )
2720, 26bitrd 187 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  ( ( N  +  1 )  -  1 )  < 
k ) )
2814, 27mpbid 146 . . . 4  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  -.  (
( N  +  1 )  -  1 )  <  k )
2911, 28pm2.21dd 609 . . 3  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  (/) )
3029ssriv 3101 . 2  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  C_  (/)
31 ss0 3403 . 2  |-  ( ( ( 0 ... N
)  i^i  ( ZZ>= `  ( N  +  1
) ) )  C_  (/) 
->  ( ( 0 ... N )  i^i  ( ZZ>=
`  ( N  + 
1 ) ) )  =  (/) )
3230, 31ax-mp 5 1  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1331    e. wcel 1480    i^i cin 3070    C_ wss 3071   (/)c0 3363   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7625   0cc0 7627   1c1 7628    + caddc 7630    < clt 7807    <_ cle 7808    - cmin 7940   ZZcz 9061   ZZ>=cuz 9333   ...cfz 9797
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798
This theorem is referenced by: (None)
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