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Theorem ordequn 4492
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 4488 . 2  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3346 . . . . 5  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eqeq2 2293 . . . . 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 3349 . . . . 5  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eqeq2 2293 . . . . 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 1628    u. cun 3151    C_ wss 3153   Ord word 4390
This theorem is referenced by:  ordun  4493  inar1  8392
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-tr 4115  df-eprel 4304  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394
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