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Theorem uzn0 11093
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 11081 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5729 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
3 fvelrnb 5913 . . 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 11092 . . . . 5  |-  ( k  e.  ZZ  ->  k  e.  ( ZZ>= `  k )
)
6 ne0i 3791 . . . . 5  |-  ( k  e.  ( ZZ>= `  k
)  ->  ( ZZ>= `  k )  =/=  (/) )
75, 6syl 16 . . . 4  |-  ( k  e.  ZZ  ->  ( ZZ>=
`  k )  =/=  (/) )
8 neeq1 2748 . . . 4  |-  ( (
ZZ>= `  k )  =  M  ->  ( ( ZZ>=
`  k )  =/=  (/) 
<->  M  =/=  (/) ) )
97, 8syl5ibcom 220 . . 3  |-  ( k  e.  ZZ  ->  (
( ZZ>= `  k )  =  M  ->  M  =/=  (/) ) )
109rexlimiv 2949 . 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 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815   (/)c0 3785   ~Pcpw 4010   ran crn 5000    Fn wfn 5581   -->wf 5582   ` cfv 5586   ZZcz 10860   ZZ>=cuz 11078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-neg 9804  df-z 10861  df-uz 11079
This theorem is referenced by:  heibor1lem  29906
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