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Theorem ordssun 4683
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 4680 . . 3  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3519 . . . . . 6  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 sseq2 3372 . . . . . 6  |-  ( ( B  u.  C )  =  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
42, 3sylbi 189 . . . . 5  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
5 olc 375 . . . . 5  |-  ( A 
C_  C  ->  ( A  C_  B  \/  A  C_  C ) )
64, 5syl6bi 221 . . . 4  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
7 ssequn2 3522 . . . . . 6  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 sseq2 3372 . . . . . 6  |-  ( ( B  u.  C )  =  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
97, 8sylbi 189 . . . . 5  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
10 orc 376 . . . . 5  |-  ( A 
C_  B  ->  ( A  C_  B  \/  A  C_  C ) )
119, 10syl6bi 221 . . . 4  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
126, 11jaoi 370 . . 3  |-  ( ( B  C_  C  \/  C  C_  B )  -> 
( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
131, 12syl 16 . 2  |-  ( ( Ord  B  /\  Ord  C )  ->  ( A  C_  ( B  u.  C
)  ->  ( A  C_  B  \/  A  C_  C ) ) )
14 ssun 3528 . 2  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
1513, 14impbid1 196 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 178    \/ wo 359    /\ wa 360    = wceq 1653    u. cun 3320    C_ wss 3322   Ord word 4582
This theorem is referenced by:  ordsucun  4807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586
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