ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fznlem Unicode version

Theorem fznlem 9007
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 8306 . . . . . . . . . . 11  |-  ( M  e.  ZZ  ->  M  e.  RR )
2 zre 8306 . . . . . . . . . . 11  |-  ( N  e.  ZZ  ->  N  e.  RR )
3 lenlt 7153 . . . . . . . . . . 11  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
41, 2, 3syl2an 277 . . . . . . . . . 10  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  <->  -.  N  <  M ) )
54biimpd 136 . . . . . . . . 9  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  <_  N  ->  -.  N  <  M
) )
65con2d 564 . . . . . . . 8  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  -.  M  <_  N
) )
76imp 119 . . . . . . 7  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  -.  M  <_  N )
87adantr 265 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  M  <_  N )
9 simplll 493 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  ZZ )
109zred 8419 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  M  e.  RR )
11 simpr 107 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  ZZ )
1211zred 8419 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  k  e.  RR )
13 simpllr 494 . . . . . . . 8  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  ZZ )
1413zred 8419 . . . . . . 7  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  N  e.  RR )
15 letr 7160 . . . . . . 7  |-  ( ( M  e.  RR  /\  k  e.  RR  /\  N  e.  RR )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
1610, 12, 14, 15syl3anc 1146 . . . . . 6  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  (
( M  <_  k  /\  k  <_  N )  ->  M  <_  N
) )
178, 16mtod 599 . . . . 5  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M )  /\  k  e.  ZZ )  ->  -.  ( M  <_  k  /\  k  <_  N ) )
1817ralrimiva 2409 . . . 4  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
19 rabeq0 3275 . . . 4  |-  ( { k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/)  <->  A. k  e.  ZZ  -.  ( M  <_  k  /\  k  <_  N ) )
2018, 19sylibr 141 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  =  (/) )
21 fzval 8978 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
2221eqeq1d 2064 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( M ... N )  =  (/)  <->  {
k  e.  ZZ  | 
( M  <_  k  /\  k  <_  N ) }  =  (/) ) )
2322adantr 265 . . 3  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( ( M ... N )  =  (/) 
<->  { k  e.  ZZ  |  ( M  <_ 
k  /\  k  <_  N ) }  =  (/) ) )
2420, 23mpbird 160 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  N  <  M
)  ->  ( M ... N )  =  (/) )
2524ex 112 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  M  ->  ( M ... N
)  =  (/) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 101    <-> wb 102    = wceq 1259    e. wcel 1409   A.wral 2323   {crab 2327   (/)c0 3252   class class class wbr 3792  (class class class)co 5540   RRcr 6946    < clt 7119    <_ cle 7120   ZZcz 8302   ...cfz 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034  ax-pre-ltwlin 7055
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2788  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-iota 4895  df-fun 4932  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-neg 7248  df-z 8303  df-fz 8977
This theorem is referenced by:  fzn  9008
  Copyright terms: Public domain W3C validator