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Theorem 0ellim 4940
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 4936 . . . 4  |-  -.  Lim  (/)
2 limeq 4890 . . . 4  |-  ( A  =  (/)  ->  ( Lim 
A  <->  Lim  (/) ) )
31, 2mtbiri 303 . . 3  |-  ( A  =  (/)  ->  -.  Lim  A )
43necon2ai 2702 . 2  |-  ( Lim 
A  ->  A  =/=  (/) )
5 limord 4937 . . 3  |-  ( Lim 
A  ->  Ord  A )
6 ord0eln0 4932 . . 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 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   Ord word 4877   Lim wlim 4879
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-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
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-pw 4012  df-sn 4028  df-pr 4030  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-lim 4883
This theorem is referenced by:  limuni3  6666  peano1  6698  oe1m  7194  oalimcl  7209  oaass  7210  oarec  7211  omlimcl  7227  odi  7228  oen0  7235  oewordri  7241  oelim2  7244  oeoalem  7245  oeoelem  7247  limensuci  7693  rankxplim2  8297  rankxplim3  8298  r1limwun  9113
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