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Theorem dflim4 4639
Description: An alternate definition of a limit ordinal. (Contributed by NM, 1-Feb-2005.)
Assertion
Ref Expression
dflim4  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
Distinct variable group:    x, A

Proof of Theorem dflim4
StepHypRef Expression
1 dflim2 4448 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A ) )
2 ordunisuc2 4635 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  A. x  e.  A  suc  x  e.  A ) )
32anbi2d 684 . . . 4  |-  ( Ord 
A  ->  ( ( (/) 
e.  A  /\  A  =  U. A )  <->  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A ) ) )
43pm5.32i 618 . . 3  |-  ( ( Ord  A  /\  ( (/) 
e.  A  /\  A  =  U. A ) )  <-> 
( Ord  A  /\  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
) )
5 3anass 938 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  ( (/)  e.  A  /\  A  =  U. A ) ) )
6 3anass 938 . . 3  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  <-> 
( Ord  A  /\  ( (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
) )
74, 5, 63bitr4i 268 . 2  |-  ( ( Ord  A  /\  (/)  e.  A  /\  A  =  U. A )  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
81, 7bitri 240 1  |-  ( Lim 
A  <->  ( Ord  A  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   (/)c0 3455   U.cuni 3827   Ord word 4391   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  limsuc  4640  limuni3  4643  oelimcl  6598
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398
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