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Theorem ordssun 4508
Description: Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)
Assertion
Ref Expression
ordssun  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  C_  ( B  u.  C
)  <->  ( A  C_  B  \/  A  C_  C
) ) )

Proof of Theorem ordssun
StepHypRef Expression
1 ordtri2or2 4505 . . 3  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3358 . . . . . 6  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 sseq2 3213 . . . . . 6  |-  ( ( B  u.  C )  =  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
42, 3sylbi 187 . . . . 5  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
5 olc 373 . . . . 5  |-  ( A 
C_  C  ->  ( A  C_  B  \/  A  C_  C ) )
64, 5syl6bi 219 . . . 4  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
7 ssequn2 3361 . . . . . 6  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 sseq2 3213 . . . . . 6  |-  ( ( B  u.  C )  =  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
97, 8sylbi 187 . . . . 5  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
10 orc 374 . . . . 5  |-  ( A 
C_  B  ->  ( A  C_  B  \/  A  C_  C ) )
119, 10syl6bi 219 . . . 4  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
126, 11jaoi 368 . . 3  |-  ( ( B  C_  C  \/  C  C_  B )  -> 
( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
131, 12syl 15 . 2  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  C_  ( B  u.  C
)  ->  ( A  C_  B  \/  A  C_  C ) ) )
14 ssun 3367 . 2  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
1513, 14impbid1 194 1  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  C_  ( B  u.  C
)  <->  ( A  C_  B  \/  A  C_  C
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1632    u. cun 3163    C_ wss 3165   Ord word 4407
This theorem is referenced by:  ordsucun  4632
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411
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