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Theorem ordun 5543
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 2462 . . 3  |-  ( A  u.  B )  =  ( A  u.  B
)
2 ordequn 5542 . . 3  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  ( A  u.  B )  ->  (
( A  u.  B
)  =  A  \/  ( A  u.  B
)  =  B ) ) )
31, 2mpi 20 . 2  |-  ( ( Ord  A  /\  Ord  B )  ->  ( ( A  u.  B )  =  A  \/  ( A  u.  B )  =  B ) )
4 ordeq 5449 . . . 4  |-  ( ( A  u.  B )  =  A  ->  ( Ord  ( A  u.  B
)  <->  Ord  A ) )
54biimprcd 233 . . 3  |-  ( Ord 
A  ->  ( ( A  u.  B )  =  A  ->  Ord  ( A  u.  B )
) )
6 ordeq 5449 . . . 4  |-  ( ( A  u.  B )  =  B  ->  ( Ord  ( A  u.  B
)  <->  Ord  B ) )
76biimprcd 233 . . 3  |-  ( Ord 
B  ->  ( ( A  u.  B )  =  B  ->  Ord  ( A  u.  B )
) )
85, 7jaao 516 . 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 374    /\ wa 375    = wceq 1455    u. cun 3414   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-ord 5445
This theorem is referenced by:  ordsucun  6679  r0weon  8469
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