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Theorem uzn0 8927
Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
Assertion
Ref Expression
uzn0  |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )

Proof of Theorem uzn0
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 uzf 8915 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5112 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
3 fvelrnb 5295 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( M  e.  ran  ZZ>= 
<->  E. k  e.  ZZ  ( ZZ>= `  k )  =  M ) )
41, 2, 3mp2b 8 . 2  |-  ( M  e.  ran  ZZ>=  <->  E. k  e.  ZZ  ( ZZ>= `  k
)  =  M )
5 uzid 8926 . . . . 5  |-  ( k  e.  ZZ  ->  k  e.  ( ZZ>= `  k )
)
6 ne0i 3275 . . . . 5  |-  ( k  e.  ( ZZ>= `  k
)  ->  ( ZZ>= `  k )  =/=  (/) )
75, 6syl 14 . . . 4  |-  ( k  e.  ZZ  ->  ( ZZ>=
`  k )  =/=  (/) )
8 neeq1 2262 . . . 4  |-  ( (
ZZ>= `  k )  =  M  ->  ( ( ZZ>=
`  k )  =/=  (/) 
<->  M  =/=  (/) ) )
97, 8syl5ibcom 153 . . 3  |-  ( k  e.  ZZ  ->  (
( ZZ>= `  k )  =  M  ->  M  =/=  (/) ) )
109rexlimiv 2477 . 2  |-  ( E. k  e.  ZZ  ( ZZ>=
`  k )  =  M  ->  M  =/=  (/) )
114, 10sylbi 119 1  |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1285    e. wcel 1434    =/= wne 2249   E.wrex 2354   (/)c0 3269   ~Pcpw 3406   ran crn 4400    Fn wfn 4962   -->wf 4963   ` cfv 4967   ZZcz 8644   ZZ>=cuz 8912
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-cnex 7337  ax-resscn 7338  ax-pre-ltirr 7358
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-res 4411  df-ima 4412  df-iota 4932  df-fun 4969  df-fn 4970  df-f 4971  df-fv 4975  df-ov 5592  df-pnf 7425  df-mnf 7426  df-xr 7427  df-ltxr 7428  df-le 7429  df-neg 7557  df-z 8645  df-uz 8913
This theorem is referenced by: (None)
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