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Theorem ordequn 5523
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 5519 . 2  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
2 ssequn1 3604 . . . . 5  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
3 eqeq2 2462 . . . . 5  |-  ( ( B  u.  C )  =  C  ->  ( A  =  ( B  u.  C )  <->  A  =  C ) )
42, 3sylbi 199 . . . 4  |-  ( B 
C_  C  ->  ( A  =  ( B  u.  C )  <->  A  =  C ) )
5 olc 386 . . . 4  |-  ( A  =  C  ->  ( A  =  B  \/  A  =  C )
)
64, 5syl6bi 232 . . 3  |-  ( B 
C_  C  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )
7 ssequn2 3607 . . . . 5  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
8 eqeq2 2462 . . . . 5  |-  ( ( B  u.  C )  =  B  ->  ( A  =  ( B  u.  C )  <->  A  =  B ) )
97, 8sylbi 199 . . . 4  |-  ( C 
C_  B  ->  ( A  =  ( B  u.  C )  <->  A  =  B ) )
10 orc 387 . . . 4  |-  ( A  =  B  ->  ( A  =  B  \/  A  =  C )
)
119, 10syl6bi 232 . . 3  |-  ( C 
C_  B  ->  ( A  =  ( B  u.  C )  ->  ( A  =  B  \/  A  =  C )
) )
126, 11jaoi 381 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    u. cun 3402    C_ wss 3404   Ord word 5422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426
This theorem is referenced by:  ordun  5524  inar1  9200
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