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Theorem fznlem 9943
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 9171 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 zre 9171 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  RR )
3 lenlt 7953 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
41, 2, 3syl2an 287 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  -.  N  <  M ) )
54biimpd 143 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  ->  -.  N  <  M
) )
65con2d 614 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  -.  M  <_  N
) )
76imp 123 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  -.  M  <_  N )
87adantr 274 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  M  <_  N )
9 simplll 523 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  ZZ )
109zred 9286 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  RR )
11 simpr 109 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  ZZ )
1211zred 9286 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  RR )
13 simpllr 524 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  ZZ )
1413zred 9286 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  RR )
15 letr 7960 . . . . . . 7  |-  ( ( M  e.  RR  /\  k  e.  RR  /\  N  e.  RR )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
1610, 12, 14, 15syl3anc 1220 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
178, 16mtod 653 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  ( M  <_  k  /\  k  <_  N ) )
1817ralrimiva 2530 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
19 rabeq0 3423 . . . 4  |-  ( { k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/)  <->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
2018, 19sylibr 133 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  =  (/) )
21 fzval 9914 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2221eqeq1d 2166 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M ... N )  =  (/)  <->  {
k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/) ) )
2322adantr 274 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( ( M ... N )  =  (/) 
<->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  (/) ) )
2420, 23mpbird 166 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( M ... N )  =  (/) )
2524ex 114 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  ( M ... N
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335    e. wcel 2128   A.wral 2435   {crab 2439   (/)c0 3394   class class class wbr 3965  (class class class)co 5824   RRcr 7731    < clt 7912    <_ cle 7913   ZZcz 9167   ...cfz 9912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltwlin 7845
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-neg 8049  df-z 9168  df-fz 9913
This theorem is referenced by:  fzn  9944
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