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Theorem oacl 6488
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
StepHypRef Expression
1 oveq2 5786 . . . 4  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
21eleq1d 2322 . . 3  |-  ( x  =  (/)  ->  ( ( A  +o  x )  e.  On  <->  ( A  +o  (/) )  e.  On ) )
3 oveq2 5786 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
43eleq1d 2322 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  y )  e.  On ) )
5 oveq2 5786 . . . 4  |-  ( x  =  suc  y  -> 
( A  +o  x
)  =  ( A  +o  suc  y ) )
65eleq1d 2322 . . 3  |-  ( x  =  suc  y  -> 
( ( A  +o  x )  e.  On  <->  ( A  +o  suc  y
)  e.  On ) )
7 oveq2 5786 . . . 4  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
87eleq1d 2322 . . 3  |-  ( x  =  B  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  B )  e.  On ) )
9 oa0 6469 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
109eleq1d 2322 . . . 4  |-  ( A  e.  On  ->  (
( A  +o  (/) )  e.  On  <->  A  e.  On ) )
1110ibir 235 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  e.  On )
12 suceloni 4562 . . . . 5  |-  ( ( A  +o  y )  e.  On  ->  suc  ( A  +o  y
)  e.  On )
13 oasuc 6477 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  +o  suc  y )  =  suc  ( A  +o  y
) )
1413eleq1d 2322 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  suc  y )  e.  On  <->  suc  ( A  +o  y
)  e.  On ) )
1512, 14syl5ibr 214 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  y )  e.  On  ->  ( A  +o  suc  y )  e.  On ) )
1615expcom 426 . . 3  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  +o  y
)  e.  On  ->  ( A  +o  suc  y
)  e.  On ) ) )
17 vex 2760 . . . . . 6  |-  x  e. 
_V
18 iunon 6309 . . . . . 6  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  +o  y )  e.  On )  ->  U_ y  e.  x  ( A  +o  y
)  e.  On )
1917, 18mpan 654 . . . . 5  |-  ( A. y  e.  x  ( A  +o  y )  e.  On  ->  U_ y  e.  x  ( A  +o  y )  e.  On )
20 oalim 6485 . . . . . . 7  |-  ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( A  +o  x )  =  U_ y  e.  x  ( A  +o  y ) )
2117, 20mpanr1 667 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A  +o  x )  = 
U_ y  e.  x  ( A  +o  y
) )
2221eleq1d 2322 . . . . 5  |-  ( ( A  e.  On  /\  Lim  x )  ->  (
( A  +o  x
)  e.  On  <->  U_ y  e.  x  ( A  +o  y )  e.  On ) )
2319, 22syl5ibr 214 . . . 4  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A. y  e.  x  ( A  +o  y
)  e.  On  ->  ( A  +o  x )  e.  On ) )
2423expcom 426 . . 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 4613 . 2  |-  ( B  e.  On  ->  ( A  e.  On  ->  ( A  +o  B )  e.  On ) )
2625impcom 421 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2757   (/)c0 3416   U_ciun 3865   Oncon0 4350   Lim wlim 4351   suc csuc 4352  (class class class)co 5778    +o coa 6430
This theorem is referenced by:  omcl  6489  oaord  6499  oacan  6500  oaword  6501  oawordri  6502  oawordeulem  6506  oalimcl  6512  oaass  6513  oaf1o  6515  odi  6531  omopth2  6536  oeoalem  6548  oeoa  6549  oancom  7306  cantnfvalf  7320  dfac12lem2  7724  cdanum  7779  wunexALT  8317
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-un 4470
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-recs 6342  df-rdg 6377  df-oadd 6437
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