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Theorem 0ellim 4454
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 4450 . . . 4  |-  -.  Lim  (/)
2 limeq 4404 . . . 4  |-  ( A  =  (/)  ->  ( Lim 
A  <->  Lim  (/) ) )
31, 2mtbiri 294 . . 3  |-  ( A  =  (/)  ->  -.  Lim  A )
43necon2ai 2491 . 2  |-  ( Lim 
A  ->  A  =/=  (/) )
5 limord 4451 . . 3  |-  ( Lim 
A  ->  Ord  A )
6 ord0eln0 4446 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
75, 6syl 15 . 2  |-  ( Lim 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
84, 7mpbird 223 1  |-  ( Lim 
A  ->  (/)  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   Ord word 4391   Lim wlim 4393
This theorem is referenced by:  limuni3  4643  peano1  4675  oe1m  6543  oalimcl  6558  oaass  6559  oarec  6560  omlimcl  6576  odi  6577  oen0  6584  oewordri  6590  oelim2  6593  oeoalem  6594  oeoelem  6596  limensuci  7037  rankxplim2  7550  rankxplim3  7551  r1limwun  8358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-lim 4397
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