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

Theorem ordzsl 6456
Description: An ordinal is zero, a successor ordinal, or a limit ordinal. (Contributed by NM, 1-Oct-2003.)
Assertion
Ref Expression
ordzsl  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Distinct variable group:    x, A

Proof of Theorem ordzsl
StepHypRef Expression
1 orduninsuc 6454 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
21biimprd 223 . . . . 5  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  A  =  U. A ) )
3 unizlim 4835 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
42, 3sylibd 214 . . . 4  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  Lim  A ) ) )
54orrd 378 . . 3  |-  ( Ord 
A  ->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A ) ) )
6 3orass 968 . . . 4  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/  Lim  A ) ) )
7 or12 523 . . . 4  |-  ( ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/ 
Lim  A ) )  <-> 
( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
86, 7bitri 249 . . 3  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
95, 8sylibr 212 . 2  |-  ( Ord 
A  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
10 ord0 4771 . . . 4  |-  Ord  (/)
11 ordeq 4726 . . . 4  |-  ( A  =  (/)  ->  ( Ord 
A  <->  Ord  (/) ) )
1210, 11mpbiri 233 . . 3  |-  ( A  =  (/)  ->  Ord  A
)
13 suceloni 6424 . . . . . 6  |-  ( x  e.  On  ->  suc  x  e.  On )
14 eleq1 2503 . . . . . 6  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1513, 14syl5ibr 221 . . . . 5  |-  ( A  =  suc  x  -> 
( x  e.  On  ->  A  e.  On ) )
16 eloni 4729 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
1715, 16syl6com 35 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  Ord  A ) )
1817rexlimiv 2835 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  Ord  A )
19 limord 4778 . . 3  |-  ( Lim 
A  ->  Ord  A )
2012, 18, 193jaoi 1281 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  ->  Ord  A )
219, 20impbii 188 1  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    \/ w3o 964    = wceq 1369    e. wcel 1756   E.wrex 2716   (/)c0 3637   U.cuni 4091   Ord word 4718   Oncon0 4719   Lim wlim 4720   suc csuc 4721
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 4413  ax-nul 4421  ax-pr 4531  ax-un 6372
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725
This theorem is referenced by:  onzsl  6457  tfrlem16  6852  omeulem1  7021  oaabs2  7084  rankxplim3  8088  rankxpsuc  8089  cardlim  8142  cardaleph  8259  cflim2  8432  dfrdg2  27609
  Copyright terms: Public domain W3C validator