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Theorem ordunpr 4654
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 4439 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 eloni 4439 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
3 ordtri2or2 4526 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
41, 2, 3syl2an 463 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  C_  C  \/  C  C_  B ) )
54orcomd 377 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  C_  B  \/  B  C_  C ) )
6 ssequn2 3382 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
7 ssequn1 3379 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
86, 7orbi12i 507 . . 3  |-  ( ( C  C_  B  \/  B  C_  C )  <->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
95, 8sylib 188 . 2  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
10 unexg 4558 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  _V )
11 elprg 3691 . . 3  |-  ( ( B  u.  C )  e.  _V  ->  (
( B  u.  C
)  e.  { B ,  C }  <->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) ) )
1210, 11syl 15 . 2  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  u.  C )  e.  { B ,  C }  <->  ( ( B  u.  C
)  =  B  \/  ( B  u.  C
)  =  C ) ) )
139, 12mpbird 223 1  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701   _Vcvv 2822    u. cun 3184    C_ wss 3186   {cpr 3675   Ord word 4428   Oncon0 4429
This theorem is referenced by:  ordunel  4655  r0weon  7685
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-tr 4151  df-eprel 4342  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433
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