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Theorem nnlim 6719
Description: A natural number is not a limit ordinal. (Contributed by NM, 18-Oct-1995.)
Assertion
Ref Expression
nnlim  |-  ( A  e.  om  ->  -.  Lim  A )

Proof of Theorem nnlim
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 nnord 6714 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 ordirr 5460 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
31, 2syl 17 . 2  |-  ( A  e.  om  ->  -.  A  e.  A )
4 elom 6709 . . . 4  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
54simprbi 465 . . 3  |-  ( A  e.  om  ->  A. x
( Lim  x  ->  A  e.  x ) )
6 limeq 5454 . . . . 5  |-  ( x  =  A  ->  ( Lim  x  <->  Lim  A ) )
7 eleq2 2502 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
86, 7imbi12d 321 . . . 4  |-  ( x  =  A  ->  (
( Lim  x  ->  A  e.  x )  <->  ( Lim  A  ->  A  e.  A
) ) )
98spcgv 3172 . . 3  |-  ( A  e.  om  ->  ( A. x ( Lim  x  ->  A  e.  x )  ->  ( Lim  A  ->  A  e.  A ) ) )
105, 9mpd 15 . 2  |-  ( A  e.  om  ->  ( Lim  A  ->  A  e.  A ) )
113, 10mtod 180 1  |-  ( A  e.  om  ->  -.  Lim  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1435    = wceq 1437    e. wcel 1870   Ord word 5441   Oncon0 5442   Lim wlim 5443   omcom 6706
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-om 6707
This theorem is referenced by:  omssnlim  6720  nnsuc  6723  cantnfp1lem2  8183  cantnflem1  8193  cnfcom2lem  8205
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