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Theorem eleenn 25005
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 3727 . 2  |-  ( A  e.  ( EE `  N )  ->  -.  ( EE `  N )  =  (/) )
2 ovex 6336 . . . . 5  |-  ( RR 
^m  ( 1 ... n ) )  e. 
_V
3 df-ee 25000 . . . . 5  |-  EE  =  ( n  e.  NN  |->  ( RR  ^m  (
1 ... n ) ) )
42, 3dmmpti 5717 . . . 4  |-  dom  EE  =  NN
54eleq2i 2541 . . 3  |-  ( N  e.  dom  EE  <->  N  e.  NN )
6 ndmfv 5903 . . 3  |-  ( -.  N  e.  dom  EE  ->  ( EE `  N
)  =  (/) )
75, 6sylnbir 314 . 2  |-  ( -.  N  e.  NN  ->  ( EE `  N )  =  (/) )
81, 7nsyl2 132 1  |-  ( A  e.  ( EE `  N )  ->  N  e.  NN )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   (/)c0 3722   dom cdm 4839   ` cfv 5589  (class class class)co 6308    ^m cmap 7490   RRcr 9556   1c1 9558   NNcn 10631   ...cfz 11810   EEcee 24997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ov 6311  df-ee 25000
This theorem is referenced by:  eleei  25006  eedimeq  25007  brbtwn  25008  brcgr  25009  eleesub  25020  eleesubd  25021  axsegconlem1  25026  axsegconlem8  25033  axeuclidlem  25071  brsegle  30946
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