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Theorem fznlem 9997
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 9216 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 zre 9216 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  RR )
3 lenlt 7995 . . . . . . . . . . 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 619 . . . . . . . 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 528 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  ZZ )
109zred 9334 . . . . . . 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 9334 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  RR )
13 simpllr 529 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  ZZ )
1413zred 9334 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  RR )
15 letr 8002 . . . . . . 7  |-  ( ( M  e.  RR  /\  k  e.  RR  /\  N  e.  RR )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
1610, 12, 14, 15syl3anc 1233 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
178, 16mtod 658 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  ( M  <_  k  /\  k  <_  N ) )
1817ralrimiva 2543 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
19 rabeq0 3444 . . . 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 9967 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2221eqeq1d 2179 . . . 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 1348    e. wcel 2141   A.wral 2448   {crab 2452   (/)c0 3414   class class class wbr 3989  (class class class)co 5853   RRcr 7773    < clt 7954    <_ cle 7955   ZZcz 9212   ...cfz 9965
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltwlin 7887
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-neg 8093  df-z 9213  df-fz 9966
This theorem is referenced by:  fzn  9998
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