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Theorem eleenn 24912
Description: If  A is in  ( EE `  N ), then 
N is a natural. (Contributed by Scott Fenton, 1-Jul-2013.)
Assertion
Ref Expression
eleenn  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )

Proof of Theorem eleenn
StepHypRef Expression
1 n0i 3766 . 2  |-  ( A  e.  ( EE `  N )  ->  -.  ( EE `  N )  =  (/) )
2 ovex 6329 . . . . 5  |-  ( RR 
^m  ( 1 ... n ) )  e. 
_V
3 df-ee 24907 . . . . 5  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
42, 3dmmpti 5721 . . . 4  |-  dom  EE  =  NN
54eleq2i 2500 . . 3  |-  ( N  e.  dom  EE  <->  N  e.  NN )
6 ndmfv 5901 . . 3  |-  ( -.  N  e.  dom  EE  ->  ( EE `  N
)  =  (/) )
75, 6sylnbir 308 . 2  |-  ( -.  N  e.  NN  ->  ( EE `  N )  =  (/) )
81, 7nsyl2 130 1  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1868   (/)c0 3761   dom cdm 4849   ` cfv 5597  (class class class)co 6301    ^m cmap 7476   RRcr 9538   1c1 9540   NNcn 10609   ...cfz 11784   EEcee 24904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fn 5600  df-fv 5605  df-ov 6304  df-ee 24907
This theorem is referenced by:  eleei  24913  eedimeq  24914  brbtwn  24915  brcgr  24916  eleesub  24927  eleesubd  24928  axsegconlem1  24933  axsegconlem8  24940  axeuclidlem  24978  brsegle  30867
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