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Theorem ordunel 4618
Description: The maximum of two ordinals belongs to a third if each of them do. (Contributed by NM, 18-Sep-2006.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
ordunel  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  A )

Proof of Theorem ordunel
StepHypRef Expression
1 prssi 3771 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
213adant1 973 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A
)
3 ordelon 4416 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
433adant3 975 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  B  e.  On )
5 ordelon 4416 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  C  e.  On )
653adant2 974 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  C  e.  On )
7 ordunpr 4617 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
84, 6, 7syl2anc 642 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  { B ,  C } )
92, 8sseldd 3181 1  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    e. wcel 1684    u. cun 3150    C_ wss 3152   {cpr 3641   Ord word 4391   Oncon0 4392
This theorem is referenced by:  oaabs2  6643  dffi3  7184  unwf  7482  rankelun  7544  infxpenlem  7641  cfsmolem  7896  r1limwun  8358  wunex2  8360
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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