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Theorem onun2i 4524
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 4513 . . 3  |-  Ord  B
3 on.1 . . . 4  |-  A  e.  On
43onordi 4513 . . 3  |-  Ord  A
5 ordtri2or 4504 . . 3  |-  ( ( Ord  B  /\  Ord  A )  ->  ( B  e.  A  \/  A  C_  B ) )
62, 4, 5mp2an 653 . 2  |-  ( B  e.  A  \/  A  C_  B )
73oneluni 4521 . . . 4  |-  ( B  e.  A  ->  ( A  u.  B )  =  A )
87, 3syl6eqel 2384 . . 3  |-  ( B  e.  A  ->  ( A  u.  B )  e.  On )
9 ssequn1 3358 . . . 4  |-  ( A 
C_  B  <->  ( A  u.  B )  =  B )
10 eleq1 2356 . . . . 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 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   Ord word 4407   Oncon0 4408
This theorem is referenced by:  rankunb  7538  rankelun  7560  rankelpr  7561  inar1  8413
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-tr 4130  df-eprel 4321  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412
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