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Theorem 0fz1 10829
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 5379 . . . . 5  |-  ( F  Fn  (/)  <->  F  =  (/) )
2 fndmu 5361 . . . . 5  |-  ( ( F  Fn  ( 1 ... N )  /\  F  Fn  (/) )  -> 
( 1 ... N
)  =  (/) )
31, 2sylan2br 462 . . . 4  |-  ( ( F  Fn  ( 1 ... N )  /\  F  =  (/) )  -> 
( 1 ... N
)  =  (/) )
43ex 423 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  ->  (
1 ... N )  =  (/) ) )
5 fneq2 5350 . . . . 5  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  Fn  (/) ) )
65, 1syl6bb 252 . . . 4  |-  ( ( 1 ... N )  =  (/)  ->  ( F  Fn  ( 1 ... N )  <->  F  =  (/) ) )
76biimpcd 215 . . 3  |-  ( F  Fn  ( 1 ... N )  ->  (
( 1 ... N
)  =  (/)  ->  F  =  (/) ) )
84, 7impbid 183 . 2  |-  ( F  Fn  ( 1 ... N )  ->  ( F  =  (/)  <->  ( 1 ... N )  =  (/) ) )
9 fz1n 10828 . 2  |-  ( N  e.  NN0  ->  ( ( 1 ... N )  =  (/)  <->  N  =  0
) )
108, 9sylan9bbr 681 1  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( F  =  (/) 
<->  N  =  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   (/)c0 3468    Fn wfn 5266  (class class class)co 5874   0cc0 8753   1c1 8754   NN0cn0 9981   ...cfz 10798
This theorem is referenced by:  clscnc  26113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799
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