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Theorem elfz2 11014
Description: Membership in a finite set of sequential integers. We use the fact that an operation's value is empty outside of its domain to show  M  e.  ZZ and  N  e.  ZZ. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
elfz2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )

Proof of Theorem elfz2
StepHypRef Expression
1 anass 631 . 2  |-  ( ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
2 df-3an 938 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ ) )
32anbi1i 677 . 2  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N
) ) )
4 elfz1 11012 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N ) ) )
5 3anass 940 . . . . 5  |-  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N )  <->  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )
6 ibar 491 . . . . 5  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
75, 6syl5bb 249 . . . 4  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( K  e.  ZZ  /\  M  <_  K  /\  K  <_  N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
84, 7bitrd 245 . . 3  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N )  <-> 
( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
9 fzf 11011 . . . . . . 7  |-  ... :
( ZZ  X.  ZZ )
--> ~P ZZ
109fdmi 5563 . . . . . 6  |-  dom  ...  =  ( ZZ  X.  ZZ )
1110ndmov 6198 . . . . 5  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N )  =  (/) )
1211eleq2d 2479 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  K  e.  (/) ) )
13 noel 3600 . . . . . 6  |-  -.  K  e.  (/)
1413pm2.21i 125 . . . . 5  |-  ( K  e.  (/)  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
15 simpl 444 . . . . 5  |-  ( ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) )  ->  ( M  e.  ZZ  /\  N  e.  ZZ ) )
1614, 15pm5.21ni 342 . . . 4  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  (/) 
<->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
1712, 16bitrd 245 . . 3  |-  ( -.  ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( K  e.  ( M ... N
)  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) ) )
188, 17pm2.61i 158 . 2  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ )  /\  ( K  e.  ZZ  /\  ( M  <_  K  /\  K  <_  N ) ) ) )
191, 3, 183bitr4ri 270 1  |-  ( K  e.  ( M ... N )  <->  ( ( M  e.  ZZ  /\  N  e.  ZZ  /\  K  e.  ZZ )  /\  ( M  <_  K  /\  K  <_  N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1721   (/)c0 3596   ~Pcpw 3767   class class class wbr 4180    X. cxp 4843  (class class class)co 6048    <_ cle 9085   ZZcz 10246   ...cfz 11007
This theorem is referenced by:  elfz4  11016  elfzuzb  11017  fzind2  11161  fzp1nel  25171  fprodntriv  25229  fprodeq0  25260  preduz  25422  fmul01lt1lem1  27589  fmul01lt1lem2  27590  stoweidlem3  27627  stoweidlem34  27658  stoweidlem51  27675  elfzmlbm  27985  elfzmlbp  27986  elfzelfzelfz  27989  swrdswrdlem  28018  swrdswrd  28019  swrdccatin2  28026  swrdccatin12lem3a  28029  swrdccatin12lem3c  28031  swrdccatin12lem3  28032  swrdccatin12  28034  swrdccatin12b  28035  swrdccatin12c  28036  swrdccat3  28037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-neg 9258  df-z 10247  df-fz 11008
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