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Theorem dflim2 4637
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 4586 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 ord0eln0 4635 . . . . 5  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
32anbi1d 686 . . . 4  |-  ( Ord 
A  ->  ( ( (/) 
e.  A  /\  A  =  U. A )  <->  ( A  =/=  (/)  /\  A  = 
U. A ) ) )
43pm5.32i 619 . . 3  |-  ( ( Ord  A  /\  ( (/) 
e.  A  /\  A  =  U. A ) )  <-> 
( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
5 3anass 940 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  ( (/)  e.  A  /\  A  =  U. A ) ) )
6 3anass 940 . . 3  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
74, 5, 63bitr4i 269 . 2  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
81, 7bitr4i 244 1  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   U.cuni 4015   Ord word 4580   Lim wlim 4582
This theorem is referenced by:  nlim0  4639  dflim4  4828
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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