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Theorem ordunpr 6534
Description: The maximum of two ordinals is equal to one of them. (Contributed by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
ordunpr  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )

Proof of Theorem ordunpr
StepHypRef Expression
1 eloni 4824 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 eloni 4824 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
3 ordtri2or2 4910 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
41, 2, 3syl2an 477 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  C_  C  \/  C  C_  B ) )
54orcomd 388 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  C_  B  \/  B  C_  C ) )
6 ssequn2 3624 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
7 ssequn1 3621 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
86, 7orbi12i 521 . . 3  |-  ( ( C  C_  B  \/  B  C_  C )  <->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
95, 8sylib 196 . 2  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
10 unexg 6478 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  _V )
11 elprg 3988 . . 3  |-  ( ( B  u.  C )  e.  _V  ->  (
( B  u.  C
)  e.  { B ,  C }  <->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) ) )
1210, 11syl 16 . 2  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  u.  C )  e.  { B ,  C }  <->  ( ( B  u.  C
)  =  B  \/  ( B  u.  C
)  =  C ) ) )
139, 12mpbird 232 1  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3065    u. cun 3421    C_ wss 3423   {cpr 3974   Ord word 4813   Oncon0 4814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-tr 4481  df-eprel 4727  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818
This theorem is referenced by:  ordunel  6535  r0weon  8277
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