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Theorem ordun 4979
Description: The maximum (i.e. union) of two ordinals is ordinal. Exercise 12 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordun  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  u.  B ) )

Proof of Theorem ordun
StepHypRef Expression
1 eqid 2467 . . 3  |-  ( A  u.  B )  =  ( A  u.  B
)
2 ordequn 4978 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  ( A  u.  B )  ->  (
( A  u.  B
)  =  A  \/  ( A  u.  B
)  =  B ) ) )
31, 2mpi 17 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  A  \/  ( A  u.  B )  =  B ) )
4 ordeq 4885 . . . 4  |-  ( ( A  u.  B )  =  A  ->  ( Ord  ( A  u.  B
)  <->  Ord  A ) )
54biimprcd 225 . . 3  |-  ( Ord 
A  ->  ( ( A  u.  B )  =  A  ->  Ord  ( A  u.  B )
) )
6 ordeq 4885 . . . 4  |-  ( ( A  u.  B )  =  B  ->  ( Ord  ( A  u.  B
)  <->  Ord  B ) )
76biimprcd 225 . . 3  |-  ( Ord 
B  ->  ( ( A  u.  B )  =  B  ->  Ord  ( A  u.  B )
) )
85, 7jaao 509 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( (
( A  u.  B
)  =  A  \/  ( A  u.  B
)  =  B )  ->  Ord  ( A  u.  B ) ) )
93, 8mpd 15 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  u.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    u. cun 3474   Ord word 4877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881
This theorem is referenced by:  ordsucun  6644  r0weon  8390
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