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Theorem nnlim 6697
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 6692 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 ordirr 4896 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
31, 2syl 16 . 2  |-  ( A  e.  om  ->  -.  A  e.  A )
4 elom 6687 . . . 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 4890 . . . . 5  |-  ( x  =  A  ->  ( Lim  x  <->  Lim  A ) )
7 eleq2 2540 . . . . 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 3198 . . 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 1377    = wceq 1379    e. wcel 1767   Ord word 4877   Oncon0 4878   Lim wlim 4879   omcom 6684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-om 6685
This theorem is referenced by:  omssnlim  6698  nnsuc  6701  cantnfp1lem2  8098  cantnflem1  8108  cantnfp1lem2OLD  8124  cantnflem1OLD  8131  cnfcom2lem  8145  cnfcom2lemOLD  8153
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