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Theorem dflim3 6681
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 4892 . 2  |-  ( Lim 
A  <->  ( Ord  A  /\  A  =/=  (/)  /\  A  =  U. A ) )
2 3anass 977 . 2  |-  ( ( Ord  A  /\  A  =/=  (/)  /\  A  = 
U. A )  <->  ( Ord  A  /\  ( A  =/=  (/)  /\  A  =  U. A ) ) )
3 df-ne 2654 . . . . . 6  |-  ( A  =/=  (/)  <->  -.  A  =  (/) )
43a1i 11 . . . . 5  |-  ( Ord 
A  ->  ( A  =/=  (/)  <->  -.  A  =  (/) ) )
5 orduninsuc 6677 . . . . 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 973    = wceq 1395    =/= wne 2652   E.wrex 2808   (/)c0 3793   U.cuni 4251   Ord word 4886   Oncon0 4887   Lim wlim 4888   suc csuc 4889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893
This theorem is referenced by:  nlimon  6685  tfinds  6693  oalimcl  7227  omlimcl  7245  r1wunlim  9132
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