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Theorem 0ellim 4781
Description: A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.)
Assertion
Ref Expression
0ellim  |-  ( Lim 
A  ->  (/)  e.  A
)

Proof of Theorem 0ellim
StepHypRef Expression
1 nlim0 4777 . . . 4  |-  -.  Lim  (/)
2 limeq 4731 . . . 4  |-  ( A  =  (/)  ->  ( Lim 
A  <->  Lim  (/) ) )
31, 2mtbiri 303 . . 3  |-  ( A  =  (/)  ->  -.  Lim  A )
43necon2ai 2656 . 2  |-  ( Lim 
A  ->  A  =/=  (/) )
5 limord 4778 . . 3  |-  ( Lim 
A  ->  Ord  A )
6 ord0eln0 4773 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
75, 6syl 16 . 2  |-  ( Lim 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
84, 7mpbird 232 1  |-  ( Lim 
A  ->  (/)  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2606   (/)c0 3637   Ord word 4718   Lim wlim 4720
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-lim 4724
This theorem is referenced by:  limuni3  6463  peano1  6495  oe1m  6984  oalimcl  6999  oaass  7000  oarec  7001  omlimcl  7017  odi  7018  oen0  7025  oewordri  7031  oelim2  7034  oeoalem  7035  oeoelem  7037  limensuci  7487  rankxplim2  8087  rankxplim3  8088  r1limwun  8903
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