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Theorem oacl 6420
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 5718 . . . 4  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
21eleq1d 2319 . . 3  |-  ( x  =  (/)  ->  ( ( A  +o  x )  e.  On  <->  ( A  +o  (/) )  e.  On ) )
3 oveq2 5718 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
43eleq1d 2319 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  y )  e.  On ) )
5 oveq2 5718 . . . 4  |-  ( x  =  suc  y  -> 
( A  +o  x
)  =  ( A  +o  suc  y ) )
65eleq1d 2319 . . 3  |-  ( x  =  suc  y  -> 
( ( A  +o  x )  e.  On  <->  ( A  +o  suc  y
)  e.  On ) )
7 oveq2 5718 . . . 4  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
87eleq1d 2319 . . 3  |-  ( x  =  B  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  B )  e.  On ) )
9 oa0 6401 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
109eleq1d 2319 . . . 4  |-  ( A  e.  On  ->  (
( A  +o  (/) )  e.  On  <->  A  e.  On ) )
1110ibir 235 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  e.  On )
12 suceloni 4495 . . . . 5  |-  ( ( A  +o  y )  e.  On  ->  suc  ( A  +o  y
)  e.  On )
13 oasuc 6409 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  +o  suc  y )  =  suc  ( A  +o  y
) )
1413eleq1d 2319 . . . . 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 2730 . . . . . 6  |-  x  e. 
_V
18 iunon 6241 . . . . . 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 6417 . . . . . . 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 2319 . . . . 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 4546 . 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 2509   _Vcvv 2727   (/)c0 3362   U_ciun 3803   Oncon0 4285   Lim wlim 4286   suc csuc 4287  (class class class)co 5710    +o coa 6362
This theorem is referenced by:  omcl  6421  oaord  6431  oacan  6432  oaword  6433  oawordri  6434  oawordeulem  6438  oalimcl  6444  oaass  6445  oaf1o  6447  odi  6463  omopth2  6468  oeoalem  6480  oeoa  6481  oancom  7236  cantnfvalf  7250  dfac12lem2  7654  cdanum  7709  wunexALT  8243
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-recs 6274  df-rdg 6309  df-oadd 6369
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