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Theorem ordunel 4810
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 3956 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
213adant1 976 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A
)
3 ordelon 4608 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
433adant3 978 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  B  e.  On )
5 ordelon 4608 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  C  e.  On )
653adant2 977 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  C  e.  On )
7 ordunpr 4809 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
84, 6, 7syl2anc 644 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  { B ,  C } )
92, 8sseldd 3351 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 937    e. wcel 1726    u. cun 3320    C_ wss 3322   {cpr 3817   Ord word 4583   Oncon0 4584
This theorem is referenced by:  oaabs2  6891  dffi3  7439  unwf  7739  rankelun  7801  infxpenlem  7900  cfsmolem  8155  r1limwun  8616  wunex2  8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
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 4216  df-opab 4270  df-tr 4306  df-eprel 4497  df-po 4506  df-so 4507  df-fr 4544  df-we 4546  df-ord 4587  df-on 4588
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