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Theorem nnlim 4858
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 4853 . . 3  |-  ( A  e.  om  ->  Ord  A )
2 ordirr 4599 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
31, 2syl 16 . 2  |-  ( A  e.  om  ->  -.  A  e.  A )
4 elom 4848 . . . 4  |-  ( A  e.  om  <->  ( A  e.  On  /\  A. x
( Lim  x  ->  A  e.  x ) ) )
54simprbi 451 . . 3  |-  ( A  e.  om  ->  A. x
( Lim  x  ->  A  e.  x ) )
6 limeq 4593 . . . . 5  |-  ( x  =  A  ->  ( Lim  x  <->  Lim  A ) )
7 eleq2 2497 . . . . 5  |-  ( x  =  A  ->  ( A  e.  x  <->  A  e.  A ) )
86, 7imbi12d 312 . . . 4  |-  ( x  =  A  ->  (
( Lim  x  ->  A  e.  x )  <->  ( Lim  A  ->  A  e.  A
) ) )
98spcgv 3036 . . 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 170 1  |-  ( A  e.  om  ->  -.  Lim  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549    = wceq 1652    e. wcel 1725   Ord word 4580   Oncon0 4581   Lim wlim 4582   omcom 4845
This theorem is referenced by:  omssnlim  4859  nnsuc  4862  cantnfp1lem2  7635  cantnflem1  7645  cnfcom2lem  7658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-tr 4303  df-eprel 4494  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846
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