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Theorem dflim2 4882
Description: An alternate definition of a limit ordinal. (Contributed by NM, 4-Nov-2004.)
Assertion
Ref Expression
dflim2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )

Proof of Theorem dflim2
StepHypRef Expression
1 df-lim 4831 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 ord0eln0 4880 . . . . 5  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
32anbi1d 704 . . . 4  |-  ( Ord 
A  ->  ( ( (/) 
e.  A  /\  A  =  U. A )  <->  ( A  =/=  (/)  /\  A  = 
U. A ) ) )
43pm5.32i 637 . . 3  |-  ( ( Ord  A  /\  ( (/) 
e.  A  /\  A  =  U. A ) )  <-> 
( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
5 3anass 969 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  ( (/)  e.  A  /\  A  =  U. A ) ) )
6 3anass 969 . . 3  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
74, 5, 63bitr4i 277 . 2  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
81, 7bitr4i 252 1  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2647   (/)c0 3744   U.cuni 4198   Ord word 4825   Lim wlim 4827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-tr 4493  df-eprel 4739  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-lim 4831
This theorem is referenced by:  nlim0  4884  dflim4  6568
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