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Theorem dflim3 4819
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 4578 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 3anass 940 . 2  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
3 df-ne 2600 . . . . . 6  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
43a1i 11 . . . . 5  |-  ( Ord 
A  ->  ( A  =/=  (/)  <->  -.  A  =  (/) ) )
5 orduninsuc 4815 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
64, 5anbi12d 692 . . . 4  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  ( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) ) )
7 ioran 477 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
86, 7syl6bbr 255 . . 3  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
98pm5.32i 619 . 2  |-  ( ( Ord  A  /\  ( A  =/=  (/)  /\  A  = 
U. A ) )  <-> 
( Ord  A  /\  -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x ) ) )
101, 2, 93bitri 263 1  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936    = wceq 1652    =/= wne 2598   E.wrex 2698   (/)c0 3620   U.cuni 4007   Ord word 4572   Oncon0 4573   Lim wlim 4574   suc csuc 4575
This theorem is referenced by:  nlimon  4823  tfinds  4831  oalimcl  6795  omlimcl  6813  r1wunlim  8604
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579
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