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Theorem eluzfz 11453
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 11452 . 2  |-  ( K  e.  ( M ... N )  <->  ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>=
`  K ) ) )
21biimpri 206 1  |-  ( ( K  e.  ( ZZ>= `  M )  /\  N  e.  ( ZZ>= `  K )
)  ->  K  e.  ( M ... N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   ` cfv 5423  (class class class)co 6096   ZZ>=cuz 10866   ...cfz 11442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-1st 6582  df-2nd 6583  df-neg 9603  df-z 10652  df-uz 10867  df-fz 11443
This theorem is referenced by:  eluzfz1  11463  eluzfz2  11464  elfz1end  11484  4fvwrd4  11538  ssnnssfz  26081  esumpmono  26533  esumcvg  26540  ssnn0ssfz  30746
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