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Theorem eleenn 25739
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 3593 . 2  |-  ( A  e.  ( EE `  N )  ->  -.  ( EE `  N )  =  (/) )
2 ovex 6065 . . . . 5  |-  ( RR 
^m  ( 1 ... n ) )  e. 
_V
3 df-ee 25734 . . . . 5  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
42, 3dmmpti 5533 . . . 4  |-  dom  EE  =  NN
54eleq2i 2468 . . 3  |-  ( N  e.  dom  EE  <->  N  e.  NN )
6 ndmfv 5714 . . 3  |-  ( -.  N  e.  dom  EE  ->  ( EE `  N
)  =  (/) )
75, 6sylnbir 299 . 2  |-  ( -.  N  e.  NN  ->  ( EE `  N )  =  (/) )
81, 7nsyl2 121 1  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   (/)c0 3588   dom cdm 4837   ` cfv 5413  (class class class)co 6040    ^m cmap 6977   RRcr 8945   1c1 8947   NNcn 9956   ...cfz 10999   EEcee 25731
This theorem is referenced by:  eleei  25740  eedimeq  25741  brbtwn  25742  brcgr  25743  eleesub  25754  eleesubd  25755  axsegconlem1  25760  axsegconlem8  25767  axeuclidlem  25805  brsegle  25946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-ov 6043  df-ee 25734
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