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Theorem 0ellim 4643
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 4639 . . . 4  |-  -.  Lim  (/)
2 limeq 4593 . . . 4  |-  ( A  =  (/)  ->  ( Lim 
A  <->  Lim  (/) ) )
31, 2mtbiri 295 . . 3  |-  ( A  =  (/)  ->  -.  Lim  A )
43necon2ai 2649 . 2  |-  ( Lim 
A  ->  A  =/=  (/) )
5 limord 4640 . . 3  |-  ( Lim 
A  ->  Ord  A )
6 ord0eln0 4635 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
75, 6syl 16 . 2  |-  ( Lim 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
84, 7mpbird 224 1  |-  ( Lim 
A  ->  (/)  e.  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   Ord word 4580   Lim wlim 4582
This theorem is referenced by:  limuni3  4832  peano1  4864  oe1m  6788  oalimcl  6803  oaass  6804  oarec  6805  omlimcl  6821  odi  6822  oen0  6829  oewordri  6835  oelim2  6838  oeoalem  6839  oeoelem  6841  limensuci  7283  rankxplim2  7804  rankxplim3  7805  r1limwun  8611
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-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
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-pw 3801  df-sn 3820  df-pr 3821  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-lim 4586
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