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Theorem 0fz1 9457
Description: Two ways to say a finite 1-based sequence is empty. (Contributed by Paul Chapman, 26-Oct-2012.)
Assertion
Ref Expression
0fz1  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( F  =  (/) 
<->  N  =  0 ) )

Proof of Theorem 0fz1
StepHypRef Expression
1 fn0 5133 . . . . 5  |-  ( F  Fn  (/)  <->  F  =  (/) )
2 fndmu 5115 . . . . 5  |-  ( ( F  Fn  ( 1 ... N )  /\  F  Fn  (/) )  -> 
( 1 ... N
)  =  (/) )
31, 2sylan2br 282 . . . 4  |-  ( ( F  Fn  ( 1 ... N )  /\  F  =  (/) )  -> 
( 1 ... N
)  =  (/) )
43ex 113 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  ->  (
1 ... N )  =  (/) ) )
5 fneq2 5103 . . . . 5  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  Fn  (/) ) )
65, 1syl6bb 194 . . . 4  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  =  (/) ) )
76biimpcd 157 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  (
( 1 ... N
)  =  (/)  ->  F  =  (/) ) )
84, 7impbid 127 . 2  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  <->  ( 1 ... N )  =  (/) ) )
9 fz1n 9456 . 2  |-  ( N  e.  NN0  ->  ( ( 1 ... N )  =  (/)  <->  N  =  0
) )
108, 9sylan9bbr 451 1  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( F  =  (/) 
<->  N  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   (/)c0 3286    Fn wfn 5010  (class class class)co 5652   0cc0 7348   1c1 7349   NN0cn0 8671   ...cfz 9422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-addcom 7443  ax-addass 7445  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-0id 7451  ax-rnegex 7452  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-inn 8421  df-n0 8672  df-z 8749  df-uz 9018  df-fz 9423
This theorem is referenced by: (None)
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