MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dflim3 Structured version   Unicode version

Theorem dflim3 6457
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 4723 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 3anass 969 . 2  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
3 df-ne 2607 . . . . . 6  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
43a1i 11 . . . . 5  |-  ( Ord 
A  ->  ( A  =/=  (/)  <->  -.  A  =  (/) ) )
5 orduninsuc 6453 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
64, 5anbi12d 710 . . . 4  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  ( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) ) )
7 ioran 490 . . . 4  |-  ( -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x )  <-> 
( -.  A  =  (/)  /\  -.  E. x  e.  On  A  =  suc  x ) )
86, 7syl6bbr 263 . . 3  |-  ( Ord 
A  ->  ( ( A  =/=  (/)  /\  A  = 
U. A )  <->  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
98pm5.32i 637 . 2  |-  ( ( Ord  A  /\  ( A  =/=  (/)  /\  A  = 
U. A ) )  <-> 
( Ord  A  /\  -.  ( A  =  (/)  \/ 
E. x  e.  On  A  =  suc  x ) ) )
101, 2, 93bitri 271 1  |-  ( Lim 
A  <->  ( Ord  A  /\  -.  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    =/= wne 2605   E.wrex 2715   (/)c0 3636   U.cuni 4090   Ord word 4717   Oncon0 4718   Lim wlim 4719   suc csuc 4720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-pss 3343  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-tp 3881  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-tr 4385  df-eprel 4631  df-po 4640  df-so 4641  df-fr 4678  df-we 4680  df-ord 4721  df-on 4722  df-lim 4723  df-suc 4724
This theorem is referenced by:  nlimon  6461  tfinds  6469  oalimcl  6998  omlimcl  7016  r1wunlim  8903
  Copyright terms: Public domain W3C validator