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Theorem uzn0 10864
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 10852 . . 3  |-  ZZ>= : ZZ --> ~P ZZ
2 ffn 5547 . . 3  |-  ( ZZ>= : ZZ --> ~P ZZ  ->  ZZ>=  Fn  ZZ )
3 fvelrnb 5727 . . 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 10863 . . . . 5  |-  ( k  e.  ZZ  ->  k  e.  ( ZZ>= `  k )
)
6 ne0i 3631 . . . . 5  |-  ( k  e.  ( ZZ>= `  k
)  ->  ( ZZ>= `  k )  =/=  (/) )
75, 6syl 16 . . . 4  |-  ( k  e.  ZZ  ->  ( ZZ>=
`  k )  =/=  (/) )
8 neeq1 2606 . . . 4  |-  ( (
ZZ>= `  k )  =  M  ->  ( ( ZZ>=
`  k )  =/=  (/) 
<->  M  =/=  (/) ) )
97, 8syl5ibcom 220 . . 3  |-  ( k  e.  ZZ  ->  (
( ZZ>= `  k )  =  M  ->  M  =/=  (/) ) )
109rexlimiv 2825 . 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 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   (/)c0 3625   ~Pcpw 3848   ran crn 4828    Fn wfn 5401   -->wf 5402   ` cfv 5406   ZZcz 10634   ZZ>=cuz 10849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-pre-lttri 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-neg 9586  df-z 10635  df-uz 10850
This theorem is referenced by:  heibor1lem  28552
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