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Theorem ordunpr 4617
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 4402 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 eloni 4402 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
3 ordtri2or2 4489 . . . . 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 3348 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
7 ssequn1 3345 . . . 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 4521 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  _V )
11 elprg 3657 . . 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 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    C_ wss 3152   {cpr 3641   Ord word 4391   Oncon0 4392
This theorem is referenced by:  ordunel  4618  r0weon  7640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396
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