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Theorem uzn0 8264
Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.)
Assertion
Ref Expression
uzn0  M  ran  ZZ>=  M  =/=  (/)

Proof of Theorem uzn0
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 uzf 8252 . . 3  ZZ>= : ZZ --> ~P ZZ
2 ffn 4989 . . 3  ZZ>= : ZZ --> ~P ZZ  ZZ>=  Fn  ZZ
3 fvelrnb 5164 . . 3  ZZ>=  Fn  ZZ  M  ran  ZZ>=  k  ZZ  ZZ>= `  k  M
41, 2, 3mp2b 8 . 2  M  ran  ZZ>=  k  ZZ  ZZ>= `  k  M
5 uzid 8263 . . . . 5  k  ZZ  k  ZZ>= `  k
6 ne0i 3224 . . . . 5  k  ZZ>= `  k  ZZ>= `  k  =/=  (/)
75, 6syl 14 . . . 4  k  ZZ  ZZ>=
`  k  =/=  (/)
8 neeq1 2213 . . . 4 
ZZ>= `  k  M 
ZZ>= `  k  =/=  (/)  M  =/=  (/)
97, 8syl5ibcom 144 . . 3  k  ZZ  ZZ>= `  k  M  M  =/=  (/)
109rexlimiv 2421 . 2  k  ZZ  ZZ>=
`  k  M  M  =/=  (/)
114, 10sylbi 114 1  M  ran  ZZ>=  M  =/=  (/)
Colors of variables: wff set class
Syntax hints:   wi 4   wb 98   wceq 1242   wcel 1390    =/= wne 2201  wrex 2301   (/)c0 3218   ~Pcpw 3351   ran crn 4289    Fn wfn 4840   -->wf 4841   ` cfv 4845   ZZcz 8021   ZZ>=cuz 8249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-pre-ltirr 6795
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-ov 5458  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-le 6863  df-neg 6982  df-z 8022  df-uz 8250
This theorem is referenced by: (None)
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