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Theorem dflim3 4638
Description: An alternate definition of a limit ordinal, which is any ordinal that is neither zero nor a successor. (Contributed by NM, 1-Nov-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
dflim3  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
Distinct variable group:    x, A

Proof of Theorem dflim3
StepHypRef Expression
1 df-lim 4397 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 3anass 938 . 2  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
3 df-ne 2448 . . . . . 6  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
43a1i 10 . . . . 5  |-  ( Ord 
A  ->  ( A  =/=  (/)  <->  -.  A  =  (/) ) )
5 orduninsuc 4634 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
64, 5anbi12d 691 . . . 4  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  ( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) ) )
7 ioran 476 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
86, 7syl6bbr 254 . . 3  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
98pm5.32i 618 . 2  |-  ( ( Ord  A  /\  ( A  =/=  (/)  /\  A  = 
U. A ) )  <-> 
( Ord  A  /\  -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x ) ) )
101, 2, 93bitri 262 1  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    =/= wne 2446   E.wrex 2544   (/)c0 3455   U.cuni 3827   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394
This theorem is referenced by:  nlimon  4642  tfinds  4650  oalimcl  6558  omlimcl  6576  r1wunlim  8359
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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