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Theorem fznlem 9455
Description: A finite set of sequential integers is empty if the bounds are reversed. (Contributed by Jim Kingdon, 16-Apr-2020.)
Assertion
Ref Expression
fznlem  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  ( M ... N
)  =  (/) ) )

Proof of Theorem fznlem
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 zre 8754 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 zre 8754 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  RR )
3 lenlt 7561 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
41, 2, 3syl2an 283 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  -.  N  <  M ) )
54biimpd 142 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  ->  -.  N  <  M
) )
65con2d 589 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  -.  M  <_  N
) )
76imp 122 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  -.  M  <_  N )
87adantr 270 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  M  <_  N )
9 simplll 500 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  ZZ )
109zred 8868 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  RR )
11 simpr 108 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  ZZ )
1211zred 8868 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  RR )
13 simpllr 501 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  ZZ )
1413zred 8868 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  RR )
15 letr 7568 . . . . . . 7  |-  ( ( M  e.  RR  /\  k  e.  RR  /\  N  e.  RR )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
1610, 12, 14, 15syl3anc 1174 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
178, 16mtod 624 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  ( M  <_  k  /\  k  <_  N ) )
1817ralrimiva 2446 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
19 rabeq0 3312 . . . 4  |-  ( { k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/)  <->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
2018, 19sylibr 132 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  =  (/) )
21 fzval 9426 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2221eqeq1d 2096 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M ... N )  =  (/)  <->  {
k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/) ) )
2322adantr 270 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( ( M ... N )  =  (/) 
<->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  (/) ) )
2420, 23mpbird 165 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( M ... N )  =  (/) )
2524ex 113 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  ( M ... N
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   A.wral 2359   {crab 2363   (/)c0 3286   class class class wbr 3845  (class class class)co 5652   RRcr 7349    < clt 7522    <_ cle 7523   ZZcz 8750   ...cfz 9424
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7436  ax-resscn 7437  ax-pre-ltwlin 7458
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7524  df-mnf 7525  df-xr 7526  df-ltxr 7527  df-le 7528  df-neg 7656  df-z 8751  df-fz 9425
This theorem is referenced by:  fzn  9456
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