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Theorem oacl 6529
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 5827 . . . 4  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
21eleq1d 2350 . . 3  |-  ( x  =  (/)  ->  ( ( A  +o  x )  e.  On  <->  ( A  +o  (/) )  e.  On ) )
3 oveq2 5827 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
43eleq1d 2350 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  y )  e.  On ) )
5 oveq2 5827 . . . 4  |-  ( x  =  suc  y  -> 
( A  +o  x
)  =  ( A  +o  suc  y ) )
65eleq1d 2350 . . 3  |-  ( x  =  suc  y  -> 
( ( A  +o  x )  e.  On  <->  ( A  +o  suc  y
)  e.  On ) )
7 oveq2 5827 . . . 4  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
87eleq1d 2350 . . 3  |-  ( x  =  B  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  B )  e.  On ) )
9 oa0 6510 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
109eleq1d 2350 . . . 4  |-  ( A  e.  On  ->  (
( A  +o  (/) )  e.  On  <->  A  e.  On ) )
1110ibir 235 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  e.  On )
12 suceloni 4603 . . . . 5  |-  ( ( A  +o  y )  e.  On  ->  suc  ( A  +o  y
)  e.  On )
13 oasuc 6518 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  +o  suc  y )  =  suc  ( A  +o  y
) )
1413eleq1d 2350 . . . . 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 2792 . . . . . 6  |-  x  e. 
_V
18 iunon 6350 . . . . . 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 6526 . . . . . . 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 2350 . . . . 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 4654 . 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 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789   (/)c0 3456   U_ciun 3906   Oncon0 4391   Lim wlim 4392   suc csuc 4393  (class class class)co 5819    +o coa 6471
This theorem is referenced by:  omcl  6530  oaord  6540  oacan  6541  oaword  6542  oawordri  6543  oawordeulem  6547  oalimcl  6553  oaass  6554  oaf1o  6556  odi  6572  omopth2  6577  oeoalem  6589  oeoa  6590  oancom  7347  cantnfvalf  7361  dfac12lem2  7765  cdanum  7820  wunexALT  8358
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-oadd 6478
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