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Theorem ordssun 4492
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 4489 . . 3  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3345 . . . . . 6  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 sseq2 3200 . . . . . 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 3348 . . . . . 6  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 sseq2 3200 . . . . . 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 3354 . 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 1623    u. cun 3150    C_ wss 3152   Ord word 4391
This theorem is referenced by:  ordsucun  4616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395
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