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Theorem uzn0 10986
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 10974 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5666 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
3 fvelrnb 5847 . . 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 10985 . . . . 5  |-  ( k  e.  ZZ  ->  k  e.  ( ZZ>= `  k )
)
6 ne0i 3750 . . . . 5  |-  ( k  e.  ( ZZ>= `  k
)  ->  ( ZZ>= `  k )  =/=  (/) )
75, 6syl 16 . . . 4  |-  ( k  e.  ZZ  ->  ( ZZ>=
`  k )  =/=  (/) )
8 neeq1 2732 . . . 4  |-  ( (
ZZ>= `  k )  =  M  ->  ( ( ZZ>=
`  k )  =/=  (/) 
<->  M  =/=  (/) ) )
97, 8syl5ibcom 220 . . 3  |-  ( k  e.  ZZ  ->  (
( ZZ>= `  k )  =  M  ->  M  =/=  (/) ) )
109rexlimiv 2939 . 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 1370    e. wcel 1758    =/= wne 2647   E.wrex 2799   (/)c0 3744   ~Pcpw 3967   ran crn 4948    Fn wfn 5520   -->wf 5521   ` cfv 5525   ZZcz 10756   ZZ>=cuz 10971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-pre-lttri 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-neg 9708  df-z 10757  df-uz 10972
This theorem is referenced by:  heibor1lem  28855
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