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Theorem ordunpr 4808
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 4593 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
2 eloni 4593 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
3 ordtri2or2 4680 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  C_  C  \/  C  C_  B ) )
41, 2, 3syl2an 465 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  C_  C  \/  C  C_  B ) )
54orcomd 379 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( C  C_  B  \/  B  C_  C ) )
6 ssequn2 3522 . . . 4  |-  ( C 
C_  B  <->  ( B  u.  C )  =  B )
7 ssequn1 3519 . . . 4  |-  ( B 
C_  C  <->  ( B  u.  C )  =  C )
86, 7orbi12i 509 . . 3  |-  ( ( C  C_  B  \/  B  C_  C )  <->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
95, 8sylib 190 . 2  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  u.  C )  =  B  \/  ( B  u.  C )  =  C ) )
10 unexg 4712 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  _V )
11 elprg 3833 . . 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 225 1  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958    u. cun 3320    C_ wss 3322   {cpr 3817   Ord word 4582   Oncon0 4583
This theorem is referenced by:  ordunel  4809  r0weon  7896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-tr 4305  df-eprel 4496  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587
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