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Theorem oacl 6550
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
Assertion
Ref Expression
oacl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )

Proof of Theorem oacl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . 4  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
21eleq1d 2362 . . 3  |-  ( x  =  (/)  ->  ( ( A  +o  x )  e.  On  <->  ( A  +o  (/) )  e.  On ) )
3 oveq2 5882 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
43eleq1d 2362 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  y )  e.  On ) )
5 oveq2 5882 . . . 4  |-  ( x  =  suc  y  -> 
( A  +o  x
)  =  ( A  +o  suc  y ) )
65eleq1d 2362 . . 3  |-  ( x  =  suc  y  -> 
( ( A  +o  x )  e.  On  <->  ( A  +o  suc  y
)  e.  On ) )
7 oveq2 5882 . . . 4  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
87eleq1d 2362 . . 3  |-  ( x  =  B  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  B )  e.  On ) )
9 oa0 6531 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
109eleq1d 2362 . . . 4  |-  ( A  e.  On  ->  (
( A  +o  (/) )  e.  On  <->  A  e.  On ) )
1110ibir 233 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  e.  On )
12 suceloni 4620 . . . . 5  |-  ( ( A  +o  y )  e.  On  ->  suc  ( A  +o  y
)  e.  On )
13 oasuc 6539 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  +o  suc  y )  =  suc  ( A  +o  y
) )
1413eleq1d 2362 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  suc  y )  e.  On  <->  suc  ( A  +o  y
)  e.  On ) )
1512, 14syl5ibr 212 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  y )  e.  On  ->  ( A  +o  suc  y )  e.  On ) )
1615expcom 424 . . 3  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  +o  y
)  e.  On  ->  ( A  +o  suc  y
)  e.  On ) ) )
17 vex 2804 . . . . . 6  |-  x  e. 
_V
18 iunon 6371 . . . . . 6  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  +o  y )  e.  On )  ->  U_ y  e.  x  ( A  +o  y
)  e.  On )
1917, 18mpan 651 . . . . 5  |-  ( A. y  e.  x  ( A  +o  y )  e.  On  ->  U_ y  e.  x  ( A  +o  y )  e.  On )
20 oalim 6547 . . . . . . 7  |-  ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( A  +o  x )  =  U_ y  e.  x  ( A  +o  y ) )
2117, 20mpanr1 664 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A  +o  x )  = 
U_ y  e.  x  ( A  +o  y
) )
2221eleq1d 2362 . . . . 5  |-  ( ( A  e.  On  /\  Lim  x )  ->  (
( A  +o  x
)  e.  On  <->  U_ y  e.  x  ( A  +o  y )  e.  On ) )
2319, 22syl5ibr 212 . . . 4  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A. y  e.  x  ( A  +o  y
)  e.  On  ->  ( A  +o  x )  e.  On ) )
2423expcom 424 . . 3  |-  ( Lim  x  ->  ( A  e.  On  ->  ( A. y  e.  x  ( A  +o  y )  e.  On  ->  ( A  +o  x )  e.  On ) ) )
252, 4, 6, 8, 11, 16, 24tfinds3 4671 . 2  |-  ( B  e.  On  ->  ( A  e.  On  ->  ( A  +o  B )  e.  On ) )
2625impcom 419 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468   U_ciun 3921   Oncon0 4408   Lim wlim 4409   suc csuc 4410  (class class class)co 5874    +o coa 6492
This theorem is referenced by:  omcl  6551  oaord  6561  oacan  6562  oaword  6563  oawordri  6564  oawordeulem  6568  oalimcl  6574  oaass  6575  oaf1o  6577  odi  6593  omopth2  6598  oeoalem  6610  oeoa  6611  oancom  7368  cantnfvalf  7382  dfac12lem2  7786  cdanum  7841  wunex3  8379
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499
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