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Theorem ordssun 4640
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 4637 . . 3  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3477 . . . . . 6  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 sseq2 3330 . . . . . 6  |-  ( ( B  u.  C )  =  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
42, 3sylbi 188 . . . . 5  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  <->  A  C_  C
) )
5 olc 374 . . . . 5  |-  ( A 
C_  C  ->  ( A  C_  B  \/  A  C_  C ) )
64, 5syl6bi 220 . . . 4  |-  ( B 
C_  C  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
7 ssequn2 3480 . . . . . 6  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 sseq2 3330 . . . . . 6  |-  ( ( B  u.  C )  =  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
97, 8sylbi 188 . . . . 5  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  <->  A  C_  B
) )
10 orc 375 . . . . 5  |-  ( A 
C_  B  ->  ( A  C_  B  \/  A  C_  C ) )
119, 10syl6bi 220 . . . 4  |-  ( C 
C_  B  ->  ( A  C_  ( B  u.  C )  ->  ( A  C_  B  \/  A  C_  C ) ) )
126, 11jaoi 369 . . 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 3486 . 2  |-  ( ( A  C_  B  \/  A  C_  C )  ->  A  C_  ( B  u.  C ) )
1513, 14impbid1 195 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 177    \/ wo 358    /\ wa 359    = wceq 1649    u. cun 3278    C_ wss 3280   Ord word 4540
This theorem is referenced by:  ordsucun  4764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544
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