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Theorem orduniorsuc 6603
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 4915 . . . . . 6  |-  ( Ord 
A  ->  U. A  C_  A )
2 orduni 6567 . . . . . . . 8  |-  ( Ord 
A  ->  Ord  U. A
)
3 ordelssne 4848 . . . . . . . 8  |-  ( ( Ord  U. A  /\  Ord  A )  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A
) ) )
42, 3mpancom 667 . . . . . . 7  |-  ( Ord 
A  ->  ( U. A  e.  A  <->  ( U. A  C_  A  /\  U. A  =/=  A ) ) )
54biimprd 223 . . . . . 6  |-  ( Ord 
A  ->  ( ( U. A  C_  A  /\  U. A  =/=  A )  ->  U. A  e.  A
) )
61, 5mpand 673 . . . . 5  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  U. A  e.  A ) )
7 ordsucss 6591 . . . . 5  |-  ( Ord 
A  ->  ( U. A  e.  A  ->  suc  U. A  C_  A ) )
86, 7syld 42 . . . 4  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  suc  U. A  C_  A )
)
9 ordsucuni 6602 . . . 4  |-  ( Ord 
A  ->  A  C_  suc  U. A )
108, 9jctild 541 . . 3  |-  ( Ord 
A  ->  ( U. A  =/=  A  ->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) ) )
11 df-ne 2600 . . . 4  |-  ( A  =/=  U. A  <->  -.  A  =  U. A )
12 necom 2672 . . . 4  |-  ( A  =/=  U. A  <->  U. A  =/= 
A )
1311, 12bitr3i 251 . . 3  |-  ( -.  A  =  U. A  <->  U. A  =/=  A )
14 eqss 3456 . . 3  |-  ( A  =  suc  U. A  <->  ( A  C_  suc  U. A  /\  suc  U. A  C_  A ) )
1510, 13, 143imtr4g 270 . 2  |-  ( Ord 
A  ->  ( -.  A  =  U. A  ->  A  =  suc  U. A
) )
1615orrd 376 1  |-  ( Ord 
A  ->  ( A  =  U. A  \/  A  =  suc  U. A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598    C_ wss 3413   U.cuni 4190   Ord word 4820   suc csuc 4823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-tr 4489  df-eprel 4733  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-ord 4824  df-on 4825  df-suc 4827
This theorem is referenced by:  onuniorsuci  6612  oeeulem  7207  cantnfp1lem2  8050  cantnflem1  8060  cantnfp1lem2OLD  8076  cantnflem1OLD  8083  cnfcom2lem  8097  cnfcom2lemOLD  8105  dfac12lem1  8475  dfac12lem2  8476  ttukeylem3  8843  ttukeylem5  8845  ttukeylem6  8846  ordtoplem  30655  ordcmp  30667  aomclem5  35347
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