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Theorem dflim3 4654
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 4413 . 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 2461 . . . . . 6  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
43a1i 10 . . . . 5  |-  ( Ord 
A  ->  ( A  =/=  (/)  <->  -.  A  =  (/) ) )
5 orduninsuc 4650 . . . . 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 1632    =/= wne 2459   E.wrex 2557   (/)c0 3468   U.cuni 3843   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410
This theorem is referenced by:  nlimon  4658  tfinds  4666  oalimcl  6574  omlimcl  6592  r1wunlim  8375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414
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