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Theorem nnlim 6488
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 6483 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 ordirr 4736 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
31, 2syl 16 . 2  |-  ( A  e.  om  ->  -.  A  e.  A )
4 elom 6478 . . . 4  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
54simprbi 464 . . 3  |-  ( A  e.  om  ->  A. x
( Lim  x  ->  A  e.  x ) )
6 limeq 4730 . . . . 5  |-  ( x  =  A  ->  ( Lim  x  <->  Lim  A ) )
7 eleq2 2503 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
86, 7imbi12d 320 . . . 4  |-  ( x  =  A  ->  (
( Lim  x  ->  A  e.  x )  <->  ( Lim  A  ->  A  e.  A
) ) )
98spcgv 3056 . . 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 177 1  |-  ( A  e.  om  ->  -.  Lim  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1367    = wceq 1369    e. wcel 1756   Ord word 4717   Oncon0 4718   Lim wlim 4719   omcom 6475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724  df-om 6476
This theorem is referenced by:  omssnlim  6489  nnsuc  6492  cantnfp1lem2  7886  cantnflem1  7896  cantnfp1lem2OLD  7912  cantnflem1OLD  7919  cnfcom2lem  7933  cnfcom2lemOLD  7941
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