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Theorem 0fz1 9776
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 5210 . . . . 5  |-  ( F  Fn  (/)  <->  F  =  (/) )
2 fndmu 5192 . . . . 5  |-  ( ( F  Fn  ( 1 ... N )  /\  F  Fn  (/) )  -> 
( 1 ... N
)  =  (/) )
31, 2sylan2br 284 . . . 4  |-  ( ( F  Fn  ( 1 ... N )  /\  F  =  (/) )  -> 
( 1 ... N
)  =  (/) )
43ex 114 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  ->  (
1 ... N )  =  (/) ) )
5 fneq2 5180 . . . . 5  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  Fn  (/) ) )
65, 1syl6bb 195 . . . 4  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  =  (/) ) )
76biimpcd 158 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  (
( 1 ... N
)  =  (/)  ->  F  =  (/) ) )
84, 7impbid 128 . 2  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  <->  ( 1 ... N )  =  (/) ) )
9 fz1n 9775 . 2  |-  ( N  e.  NN0  ->  ( ( 1 ... N )  =  (/)  <->  N  =  0
) )
108, 9sylan9bbr 456 1  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( F  =  (/) 
<->  N  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1314    e. wcel 1463   (/)c0 3331    Fn wfn 5086  (class class class)co 5740   0cc0 7584   1c1 7585   NN0cn0 8931   ...cfz 9741
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-apti 7699  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742
This theorem is referenced by: (None)
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