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Theorem ordun 4893
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 2382 . . 3  |-  ( A  u.  B )  =  ( A  u.  B
)
2 ordequn 4892 . . 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 4799 . . . 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 4799 . . . 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 507 . 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 366    /\ wa 367    = wceq 1399    u. cun 3387   Ord word 4791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-tr 4461  df-eprel 4705  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795
This theorem is referenced by:  ordsucun  6559  r0weon  8303
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