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Theorem uzn0 11107
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 11095 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5721 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
3 fvelrnb 5905 . . 3  |-  ( ZZ>=  Fn  ZZ  ->  ( M  e.  ran  ZZ>= 
<->  E. k  e.  ZZ  ( ZZ>= `  k )  =  M ) )
41, 2, 3mp2b 10 . 2  |-  ( M  e.  ran  ZZ>=  <->  E. k  e.  ZZ  ( ZZ>= `  k
)  =  M )
5 uzid 11106 . . . . 5  |-  ( k  e.  ZZ  ->  k  e.  ( ZZ>= `  k )
)
6 ne0i 3776 . . . . 5  |-  ( k  e.  ( ZZ>= `  k
)  ->  ( ZZ>= `  k )  =/=  (/) )
75, 6syl 16 . . . 4  |-  ( k  e.  ZZ  ->  ( ZZ>=
`  k )  =/=  (/) )
8 neeq1 2724 . . . 4  |-  ( (
ZZ>= `  k )  =  M  ->  ( ( ZZ>=
`  k )  =/=  (/) 
<->  M  =/=  (/) ) )
97, 8syl5ibcom 220 . . 3  |-  ( k  e.  ZZ  ->  (
( ZZ>= `  k )  =  M  ->  M  =/=  (/) ) )
109rexlimiv 2929 . 2  |-  ( E. k  e.  ZZ  ( ZZ>=
`  k )  =  M  ->  M  =/=  (/) )
114, 10sylbi 195 1  |-  ( M  e.  ran  ZZ>=  ->  M  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1383    e. wcel 1804    =/= wne 2638   E.wrex 2794   (/)c0 3770   ~Pcpw 3997   ran crn 4990    Fn wfn 5573   -->wf 5574   ` cfv 5578   ZZcz 10871   ZZ>=cuz 11092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-pre-lttri 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-ov 6284  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-neg 9813  df-z 10872  df-uz 11093
This theorem is referenced by:  heibor1lem  30281
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