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Theorem dflim2 4848
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 4797 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 ord0eln0 4846 . . . . 5  |-  ( Ord 
A  ->  ( (/)  e.  A  <->  A  =/=  (/) ) )
32anbi1d 702 . . . 4  |-  ( Ord 
A  ->  ( ( (/) 
e.  A  /\  A  =  U. A )  <->  ( A  =/=  (/)  /\  A  = 
U. A ) ) )
43pm5.32i 635 . . 3  |-  ( ( Ord  A  /\  ( (/) 
e.  A  /\  A  =  U. A ) )  <-> 
( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
5 3anass 975 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  ( (/)  e.  A  /\  A  =  U. A ) ) )
6 3anass 975 . . 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 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   (/)c0 3711   U.cuni 4163   Ord word 4791   Lim wlim 4793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-lim 4797
This theorem is referenced by:  nlim0  4850  dflim4  6582
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