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Theorem nn0disj 9698
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 3198 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  <->  ( k  e.  ( 0 ... N
)  /\  k  e.  ( ZZ>= `  ( N  +  1 ) ) ) )
21simprbi 270 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( ZZ>= `  ( N  +  1 ) ) )
3 eluzle 9130 . . . . . 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 9123 . . . . . . 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 9127 . . . . . . 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 8904 . . . . . 6  |-  ( ( ( N  +  1 )  e.  ZZ  /\  k  e.  ZZ )  ->  ( ( N  + 
1 )  <_  k  <->  ( ( N  +  1 )  -  1 )  <  k ) )
106, 8, 9syl2anc 404 . . . . 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 269 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  ( 0 ... N
) )
13 elfzle2 9591 . . . . . 6  |-  ( k  e.  ( 0 ... N )  ->  k  <_  N )
1412, 13syl 14 . . . . 5  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  <_  N )
158zred 8967 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  RR )
16 elfzel2 9587 . . . . . . . . . 10  |-  ( k  e.  ( 0 ... N )  ->  N  e.  ZZ )
1716adantr 271 . . . . . . . . 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 8967 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  RR )
2015, 19lenltd 7698 . . . . . 6  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( k  <_  N  <->  -.  N  <  k ) )
2118zcnd 8968 . . . . . . . . . 10  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  e.  CC )
22 pncan1 7952 . . . . . . . . . 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 2100 . . . . . . . 8  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  N  =  ( ( N  + 
1 )  -  1 ) )
2524breq1d 3877 . . . . . . 7  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  ( N  <  k  <->  ( ( N  +  1 )  - 
1 )  <  k
) )
2625notbid 630 . . . . . 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 588 . . 3  |-  ( k  e.  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  ->  k  e.  (/) )
3029ssriv 3043 . 2  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  C_  (/)
31 ss0 3342 . 2  |-  ( ( ( 0 ... N
)  i^i  ( ZZ>= `  ( N  +  1
) ) )  C_  (/) 
->  ( ( 0 ... N )  i^i  ( ZZ>=
`  ( N  + 
1 ) ) )  =  (/) )
3230, 31ax-mp 7 1  |-  ( ( 0 ... N )  i^i  ( ZZ>= `  ( N  +  1 ) ) )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    = wceq 1296    e. wcel 1445    i^i cin 3012    C_ wss 3013   (/)c0 3302   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   CCcc 7445   0cc0 7447   1c1 7448    + caddc 7450    < clt 7619    <_ cle 7620    - cmin 7750   ZZcz 8848   ZZ>=cuz 9118   ...cfz 9573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-addcom 7542  ax-addass 7544  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-0id 7550  ax-rnegex 7551  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-ltadd 7558
This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-inn 8521  df-n0 8772  df-z 8849  df-uz 9119  df-fz 9574
This theorem is referenced by: (None)
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