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Theorem eluzfz 10986
Description: Membership in a finite set of sequential integers. (Contributed by NM, 4-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
eluzfz  |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K )
)  ->  K  e.  ( M ... N ) )

Proof of Theorem eluzfz
StepHypRef Expression
1 elfzuzb 10985 . 2  |-  ( K  e.  ( M ... N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  K ) ) )
21biimpri 198 1  |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K )
)  ->  K  e.  ( M ... N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1717   ` cfv 5394  (class class class)co 6020   ZZ>=cuz 10420   ...cfz 10975
This theorem is referenced by:  eluzfz1  10996  eluzfz2  10997  elfz1end  11013  4fvwrd4  11051  ssnnssfz  23984  esumpmono  24265  esumcvg  24272
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-cnex 8979  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-neg 9226  df-z 10215  df-uz 10421  df-fz 10976
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