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Theorem ordequn 4451
Description: The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordequn  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )

Proof of Theorem ordequn
StepHypRef Expression
1 ordtri2or2 4447 . 2  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3306 . . . . 5  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eqeq2 2265 . . . . 5  |-  ( ( B  u.  C )  =  C  ->  ( A  =  ( B  u.  C )  <->  A  =  C ) )
42, 3sylbi 189 . . . 4  |-  ( B 
C_  C  ->  ( A  =  ( B  u.  C )  <->  A  =  C ) )
5 olc 375 . . . 4  |-  ( A  =  C  ->  ( A  =  B  \/  A  =  C )
)
64, 5syl6bi 221 . . 3  |-  ( B 
C_  C  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )
7 ssequn2 3309 . . . . 5  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eqeq2 2265 . . . . 5  |-  ( ( B  u.  C )  =  B  ->  ( A  =  ( B  u.  C )  <->  A  =  B ) )
97, 8sylbi 189 . . . 4  |-  ( C 
C_  B  ->  ( A  =  ( B  u.  C )  <->  A  =  B ) )
10 orc 376 . . . 4  |-  ( A  =  B  ->  ( A  =  B  \/  A  =  C )
)
119, 10syl6bi 221 . . 3  |-  ( C 
C_  B  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )
126, 11jaoi 370 . 2  |-  ( ( B  C_  C  \/  C  C_  B )  -> 
( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C ) ) )
131, 12syl 17 1  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1619    u. cun 3111    C_ wss 3113   Ord word 4349
This theorem is referenced by:  ordun  4452  inar1  8351
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 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172
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 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-tr 4074  df-eprel 4263  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353
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