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Theorem ordunpr 6634
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 4877 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 eloni 4877 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
3 ordtri2or2 4963 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
41, 2, 3syl2an 475 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  C_  C  \/  C  C_  B ) )
54orcomd 386 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  C_  B  \/  B  C_  C ) )
6 ssequn2 3663 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
7 ssequn1 3660 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
86, 7orbi12i 519 . . 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 6574 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  _V )
11 elprg 4032 . . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106    u. cun 3459    C_ wss 3461   {cpr 4018   Ord word 4866   Oncon0 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871
This theorem is referenced by:  ordunel  6635  r0weon  8381
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