MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onun2i Unicode version

Theorem onun2i 4508
Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)
Hypotheses
Ref Expression
on.1  |-  A  e.  On
on.2  |-  B  e.  On
Assertion
Ref Expression
onun2i  |-  ( A  u.  B )  e.  On

Proof of Theorem onun2i
StepHypRef Expression
1 on.2 . . . 4  |-  B  e.  On
21onordi 4497 . . 3  |-  Ord  B
3 on.1 . . . 4  |-  A  e.  On
43onordi 4497 . . 3  |-  Ord  A
5 ordtri2or 4488 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  \/  A  C_  B ) )
62, 4, 5mp2an 653 . 2  |-  ( B  e.  A  \/  A  C_  B )
73oneluni 4505 . . . 4  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
87, 3syl6eqel 2371 . . 3  |-  ( B  e.  A  ->  ( A  u.  B )  e.  On )
9 ssequn1 3345 . . . 4  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
10 eleq1 2343 . . . . 5  |-  ( ( A  u.  B )  =  B  ->  (
( A  u.  B
)  e.  On  <->  B  e.  On ) )
111, 10mpbiri 224 . . . 4  |-  ( ( A  u.  B )  =  B  ->  ( A  u.  B )  e.  On )
129, 11sylbi 187 . . 3  |-  ( A 
C_  B  ->  ( A  u.  B )  e.  On )
138, 12jaoi 368 . 2  |-  ( ( B  e.  A  \/  A  C_  B )  -> 
( A  u.  B
)  e.  On )
146, 13ax-mp 8 1  |-  ( A  u.  B )  e.  On
Colors of variables: wff set class
Syntax hints:    \/ wo 357    = wceq 1623    e. wcel 1684    u. cun 3150    C_ wss 3152   Ord word 4391   Oncon0 4392
This theorem is referenced by:  rankunb  7522  rankelun  7544  rankelpr  7545  inar1  8397
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-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
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
  Copyright terms: Public domain W3C validator