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Theorem fzn0 10825
Description: Properties of a finite interval of integers which is non-empty. (Contributed by Jeff Madsen, 17-Jun-2010.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
fzn0  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )

Proof of Theorem fzn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 n0 3477 . . 3  |-  ( ( M ... N )  =/=  (/)  <->  E. x  x  e.  ( M ... N
) )
2 elfzuz2 10817 . . . 4  |-  ( x  e.  ( M ... N )  ->  N  e.  ( ZZ>= `  M )
)
32exlimiv 1624 . . 3  |-  ( E. x  x  e.  ( M ... N )  ->  N  e.  (
ZZ>= `  M ) )
41, 3sylbi 187 . 2  |-  ( ( M ... N )  =/=  (/)  ->  N  e.  ( ZZ>= `  M )
)
5 eluzfz1 10819 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
6 ne0i 3474 . . 3  |-  ( M  e.  ( M ... N )  ->  ( M ... N )  =/=  (/) )
75, 6syl 15 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( M ... N )  =/=  (/) )
84, 7impbii 180 1  |-  ( ( M ... N )  =/=  (/)  <->  N  e.  ( ZZ>=
`  M ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    e. wcel 1696    =/= wne 2459   (/)c0 3468   ` cfv 5271  (class class class)co 5874   ZZ>=cuz 10246   ...cfz 10798
This theorem is referenced by:  fzn  10826  fzfi  11050  fseqsupcl  11055  fsumrev2  12260  gsumval3  15207  iscmet3  18735  dchrisum0flblem1  20673  pntrsumbnd2  20732  fsumprd  25432  stoweidlem26  27878
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-neg 9056  df-z 10041  df-uz 10247  df-fz 10799
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