Theorem 0fz1 10382

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

Step	Hyp	Ref	Expression
1		fn0 5480	. . . . 5 |- ( F Fn (/) <-> F = (/) )
2		fndmu 5461	. . . . 5 |- ( ( F Fn ( 1 ... N ) /\ F Fn (/) ) -> ( 1 ... N ) = (/) )
3	1, 2	sylan2br 288	. . . 4 |- ( ( F Fn ( 1 ... N ) /\ F = (/) ) -> ( 1 ... N ) = (/) )
4	3	ex 115	. . 3 |- ( F Fn ( 1 ... N ) -> ( F = (/) -> ( 1 ... N ) = (/) ) )
5		fneq2 5447	. . . . 5 |- ( ( 1 ... N ) = (/) -> ( F Fn ( 1 ... N ) <-> F Fn (/) ) )
6	5, 1	bitrdi 196	. . . 4 |- ( ( 1 ... N ) = (/) -> ( F Fn ( 1 ... N ) <-> F = (/) ) )
7	6	biimpcd 159	. . 3 |- ( F Fn ( 1 ... N ) -> ( ( 1 ... N ) = (/) -> F = (/) ) )
8	4, 7	impbid 129	. 2 |- ( F Fn ( 1 ... N ) -> ( F = (/) <-> ( 1 ... N ) = (/) ) )
9		fz1n 10381	. 2 |- ( N e. NN0 -> ( ( 1 ... N ) = (/) <-> N = 0 ) )
10	8, 9	sylan9bbr 463	1 |- ( ( N e. NN0 /\ F Fn ( 1 ... N ) ) -> ( F = (/) <-> N = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   (/)c0 3510   Fn wfn 5349  (class class class)co 6052   0cc0 8129   1c1 8130   NN0cn0 9498   ...cfz 10345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346

This theorem is referenced by: (None)