MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordunel Unicode version

Theorem ordunel 4770
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 3918 . . 3  |-  ( ( B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A )
213adant1 975 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  { B ,  C }  C_  A
)
3 ordelon 4569 . . . 4  |-  ( ( Ord  A  /\  B  e.  A )  ->  B  e.  On )
433adant3 977 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  B  e.  On )
5 ordelon 4569 . . . 4  |-  ( ( Ord  A  /\  C  e.  A )  ->  C  e.  On )
653adant2 976 . . 3  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  C  e.  On )
7 ordunpr 4769 . . 3  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  u.  C
)  e.  { B ,  C } )
84, 6, 7syl2anc 643 . 2  |-  ( ( Ord  A  /\  B  e.  A  /\  C  e.  A )  ->  ( B  u.  C )  e.  { B ,  C } )
92, 8sseldd 3313 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 936    e. wcel 1721    u. cun 3282    C_ wss 3284   {cpr 3779   Ord word 4544   Oncon0 4545
This theorem is referenced by:  oaabs2  6851  dffi3  7398  unwf  7696  rankelun  7758  infxpenlem  7855  cfsmolem  8110  r1limwun  8571  wunex2  8573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-tr 4267  df-eprel 4458  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549
  Copyright terms: Public domain W3C validator