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Theorem orduniorsuc 4637
Description: An ordinal class is either its union or the successor of its union. If we adopt the view that zero is a limit ordinal, this means every ordinal class is either a limit or a successor. (Contributed by NM, 13-Sep-2003.)
Assertion
Ref Expression
orduniorsuc  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )

Proof of Theorem orduniorsuc
StepHypRef Expression
1 orduniss 4503 . . . . . 6  |-  ( Ord 
A  ->  U. A  C_  A )
2 orduni 4601 . . . . . . . 8  |-  ( Ord 
A  ->  Ord  U. A
)
3 ordelssne 4435 . . . . . . . 8  |-  ( ( Ord  U. A  /\  Ord  A )  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A
) ) )
42, 3mpancom 650 . . . . . . 7  |-  ( Ord 
A  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A ) ) )
54biimprd 214 . . . . . 6  |-  ( Ord 
A  ->  ( ( U. A  C_  A  /\  U. A  =/=  A )  ->  U. A  e.  A
) )
61, 5mpand 656 . . . . 5  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  U. A  e.  A ) )
7 ordsucss 4625 . . . . 5  |-  ( Ord 
A  ->  ( U. A  e.  A  ->  suc  U. A  C_  A ) )
86, 7syld 40 . . . 4  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  suc  U. A  C_  A )
)
9 ordsucuni 4636 . . . 4  |-  ( Ord 
A  ->  A  C_  suc  U. A )
108, 9jctild 527 . . 3  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) ) )
11 df-ne 2461 . . . 4  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
12 necom 2540 . . . 4  |-  ( A  =/=  U. A  <->  U. A  =/= 
A )
1311, 12bitr3i 242 . . 3  |-  ( -.  A  =  U. A  <->  U. A  =/=  A )
14 eqss 3207 . . 3  |-  ( A  =  suc  U. A  <->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) )
1510, 13, 143imtr4g 261 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
1615orrd 367 1  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    C_ wss 3165   U.cuni 3843   Ord word 4407   suc csuc 4410
This theorem is referenced by:  onuniorsuci  4646  oeeulem  6615  cantnfp1lem2  7397  cantnflem1  7407  cnfcom2lem  7420  dfac12lem1  7785  dfac12lem2  7786  ttukeylem3  8154  ttukeylem5  8156  ttukeylem6  8157  ordtoplem  24946  ordcmp  24958  aomclem5  27258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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