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Theorem 0fz1 9947
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 5289 . . . . 5  |-  ( F  Fn  (/)  <->  F  =  (/) )
2 fndmu 5271 . . . . 5  |-  ( ( F  Fn  ( 1 ... N )  /\  F  Fn  (/) )  -> 
( 1 ... N
)  =  (/) )
31, 2sylan2br 286 . . . 4  |-  ( ( F  Fn  ( 1 ... N )  /\  F  =  (/) )  -> 
( 1 ... N
)  =  (/) )
43ex 114 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  ->  (
1 ... N )  =  (/) ) )
5 fneq2 5259 . . . . 5  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  Fn  (/) ) )
65, 1bitrdi 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 9946 . 2  |-  ( N  e.  NN0  ->  ( ( 1 ... N )  =  (/)  <->  N  =  0
) )
108, 9sylan9bbr 459 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 1335    e. wcel 2128   (/)c0 3394    Fn wfn 5165  (class class class)co 5824   0cc0 7732   1c1 7733   NN0cn0 9090   ...cfz 9912
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-addcom 7832  ax-addass 7834  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-0id 7840  ax-rnegex 7841  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-inn 8834  df-n0 9091  df-z 9168  df-uz 9440  df-fz 9913
This theorem is referenced by: (None)
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