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Theorem ordun 4385
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 2253 . . 3  |-  ( A  u.  B )  =  ( A  u.  B
)
2 ordequn 4384 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  ( A  u.  B )  ->  (
( A  u.  B
)  =  A  \/  ( A  u.  B
)  =  B ) ) )
31, 2mpi 18 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  A  \/  ( A  u.  B )  =  B ) )
4 ordeq 4292 . . . 4  |-  ( ( A  u.  B )  =  A  ->  ( Ord  ( A  u.  B
)  <->  Ord  A ) )
54biimprcd 218 . . 3  |-  ( Ord 
A  ->  ( ( A  u.  B )  =  A  ->  Ord  ( A  u.  B )
) )
6 ordeq 4292 . . . 4  |-  ( ( A  u.  B )  =  B  ->  ( Ord  ( A  u.  B
)  <->  Ord  B ) )
76biimprcd 218 . . 3  |-  ( Ord 
B  ->  ( ( A  u.  B )  =  B  ->  Ord  ( A  u.  B )
) )
85, 7jaao 497 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( (
( A  u.  B
)  =  A  \/  ( A  u.  B
)  =  B )  ->  Ord  ( A  u.  B ) ) )
93, 8mpd 16 1  |-  ( ( Ord  A  /\  Ord  B )  ->  Ord  ( A  u.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    u. cun 3076   Ord word 4284
This theorem is referenced by:  ordsucun  4507  r0weon  7524
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-tr 4011  df-eprel 4198  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288
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