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Theorem 0ellim 5504
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 5500 . . . 4  |-  -.  Lim  (/)
2 limeq 5454 . . . 4  |-  ( A  =  (/)  ->  ( Lim 
A  <->  Lim  (/) ) )
31, 2mtbiri 304 . . 3  |-  ( A  =  (/)  ->  -.  Lim  A )
43necon2ai 2666 . 2  |-  ( Lim 
A  ->  A  =/=  (/) )
5 limord 5501 . . 3  |-  ( Lim 
A  ->  Ord  A )
6 ord0eln0 5496 . . 3  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
75, 6syl 17 . 2  |-  ( Lim 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
84, 7mpbird 235 1  |-  ( Lim 
A  ->  (/)  e.  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1870    =/= wne 2625   (/)c0 3767   Ord word 5441   Lim wlim 5443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-ord 5445  df-lim 5447
This theorem is referenced by:  limuni3  6693  peano1  6726  oe1m  7254  oalimcl  7269  oaass  7270  oarec  7271  omlimcl  7287  odi  7288  oen0  7295  oewordri  7301  oelim2  7304  oeoalem  7305  oeoelem  7307  limensuci  7754  rankxplim2  8350  rankxplim3  8351  r1limwun  9160
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