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Theorem fz0 11713
Description: A finite set of sequential integers is empty if its bounds are not integers. (Contributed by AV, 13-Oct-2018.)
Assertion
Ref Expression
fz0  |-  ( ( M  e/  ZZ  \/  N  e/  ZZ )  -> 
( M ... N
)  =  (/) )

Proof of Theorem fz0
StepHypRef Expression
1 df-nel 2665 . . 3  |-  ( M  e/  ZZ  <->  -.  M  e.  ZZ )
2 df-nel 2665 . . 3  |-  ( N  e/  ZZ  <->  -.  N  e.  ZZ )
31, 2orbi12i 521 . 2  |-  ( ( M  e/  ZZ  \/  N  e/  ZZ )  <->  ( -.  M  e.  ZZ  \/  -.  N  e.  ZZ ) )
4 ianor 488 . . 3  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  <-> 
( -.  M  e.  ZZ  \/  -.  N  e.  ZZ ) )
5 fzf 11688 . . . . 5  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
65fdmi 5742 . . . 4  |-  dom  ...  =  ( ZZ  X.  ZZ )
76ndmov 6454 . . 3  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  (/) )
84, 7sylbir 213 . 2  |-  ( ( -.  M  e.  ZZ  \/  -.  N  e.  ZZ )  ->  ( M ... N )  =  (/) )
93, 8sylbi 195 1  |-  ( ( M  e/  ZZ  \/  N  e/  ZZ )  -> 
( M ... N
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    e/ wnel 2663   (/)c0 3790   ~Pcpw 4016    X. cxp 5003  (class class class)co 6295   ZZcz 10876   ...cfz 11684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-neg 9820  df-z 10877  df-fz 11685
This theorem is referenced by:  ffz0iswrd  12549
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