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Theorem ordzsl 4715
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 4713 . . . . . 6  |-  ( Ord 
A  ->  ( A  =  U. A  <->  -.  E. x  e.  On  A  =  suc  x ) )
21biimprd 214 . . . . 5  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  A  =  U. A ) )
3 unizlim 4588 . . . . 5  |-  ( Ord 
A  ->  ( A  =  U. A  <->  ( A  =  (/)  \/  Lim  A
) ) )
42, 3sylibd 205 . . . 4  |-  ( Ord 
A  ->  ( -.  E. x  e.  On  A  =  suc  x  ->  ( A  =  (/)  \/  Lim  A ) ) )
54orrd 367 . . 3  |-  ( Ord 
A  ->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A ) ) )
6 3orass 937 . . . 4  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/  Lim  A ) ) )
7 or12 509 . . . 4  |-  ( ( A  =  (/)  \/  ( E. x  e.  On  A  =  suc  x  \/ 
Lim  A ) )  <-> 
( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
86, 7bitri 240 . . 3  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  <->  ( E. x  e.  On  A  =  suc  x  \/  ( A  =  (/)  \/  Lim  A
) ) )
95, 8sylibr 203 . 2  |-  ( Ord 
A  ->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
10 ord0 4523 . . . 4  |-  Ord  (/)
11 ordeq 4478 . . . 4  |-  ( A  =  (/)  ->  ( Ord 
A  <->  Ord  (/) ) )
1210, 11mpbiri 224 . . 3  |-  ( A  =  (/)  ->  Ord  A
)
13 suceloni 4683 . . . . . 6  |-  ( x  e.  On  ->  suc  x  e.  On )
14 eleq1 2418 . . . . . 6  |-  ( A  =  suc  x  -> 
( A  e.  On  <->  suc  x  e.  On ) )
1513, 14syl5ibr 212 . . . . 5  |-  ( A  =  suc  x  -> 
( x  e.  On  ->  A  e.  On ) )
16 eloni 4481 . . . . 5  |-  ( A  e.  On  ->  Ord  A )
1715, 16syl6com 31 . . . 4  |-  ( x  e.  On  ->  ( A  =  suc  x  ->  Ord  A ) )
1817rexlimiv 2737 . . 3  |-  ( E. x  e.  On  A  =  suc  x  ->  Ord  A )
19 limord 4530 . . 3  |-  ( Lim 
A  ->  Ord  A )
2012, 18, 193jaoi 1245 . 2  |-  ( ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A )  ->  Ord  A )
219, 20impbii 180 1  |-  ( Ord 
A  <->  ( A  =  (/)  \/  E. x  e.  On  A  =  suc  x  \/  Lim  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    \/ w3o 933    = wceq 1642    e. wcel 1710   E.wrex 2620   (/)c0 3531   U.cuni 3906   Ord word 4470   Oncon0 4471   Lim wlim 4472   suc csuc 4473
This theorem is referenced by:  onzsl  4716  tfrlem16  6493  omeulem1  6664  oaabs2  6727  rankxplim3  7638  rankxpsuc  7639  cardlim  7692  cardaleph  7803  cflim2  7976  dfrdg2  24710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-tr 4193  df-eprel 4384  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477
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