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Theorem oacl 6746
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 6056 . . . 4  |-  ( x  =  (/)  ->  ( A  +o  x )  =  ( A  +o  (/) ) )
21eleq1d 2478 . . 3  |-  ( x  =  (/)  ->  ( ( A  +o  x )  e.  On  <->  ( A  +o  (/) )  e.  On ) )
3 oveq2 6056 . . . 4  |-  ( x  =  y  ->  ( A  +o  x )  =  ( A  +o  y
) )
43eleq1d 2478 . . 3  |-  ( x  =  y  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  y )  e.  On ) )
5 oveq2 6056 . . . 4  |-  ( x  =  suc  y  -> 
( A  +o  x
)  =  ( A  +o  suc  y ) )
65eleq1d 2478 . . 3  |-  ( x  =  suc  y  -> 
( ( A  +o  x )  e.  On  <->  ( A  +o  suc  y
)  e.  On ) )
7 oveq2 6056 . . . 4  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
87eleq1d 2478 . . 3  |-  ( x  =  B  ->  (
( A  +o  x
)  e.  On  <->  ( A  +o  B )  e.  On ) )
9 oa0 6727 . . . . 5  |-  ( A  e.  On  ->  ( A  +o  (/) )  =  A )
109eleq1d 2478 . . . 4  |-  ( A  e.  On  ->  (
( A  +o  (/) )  e.  On  <->  A  e.  On ) )
1110ibir 234 . . 3  |-  ( A  e.  On  ->  ( A  +o  (/) )  e.  On )
12 suceloni 4760 . . . . 5  |-  ( ( A  +o  y )  e.  On  ->  suc  ( A  +o  y
)  e.  On )
13 oasuc 6735 . . . . . 6  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  +o  suc  y )  =  suc  ( A  +o  y
) )
1413eleq1d 2478 . . . . 5  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  suc  y )  e.  On  <->  suc  ( A  +o  y
)  e.  On ) )
1512, 14syl5ibr 213 . . . 4  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  +o  y )  e.  On  ->  ( A  +o  suc  y )  e.  On ) )
1615expcom 425 . . 3  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  +o  y
)  e.  On  ->  ( A  +o  suc  y
)  e.  On ) ) )
17 vex 2927 . . . . . 6  |-  x  e. 
_V
18 iunon 6567 . . . . . 6  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  +o  y )  e.  On )  ->  U_ y  e.  x  ( A  +o  y
)  e.  On )
1917, 18mpan 652 . . . . 5  |-  ( A. y  e.  x  ( A  +o  y )  e.  On  ->  U_ y  e.  x  ( A  +o  y )  e.  On )
20 oalim 6743 . . . . . . 7  |-  ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( A  +o  x )  =  U_ y  e.  x  ( A  +o  y ) )
2117, 20mpanr1 665 . . . . . 6  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A  +o  x )  = 
U_ y  e.  x  ( A  +o  y
) )
2221eleq1d 2478 . . . . 5  |-  ( ( A  e.  On  /\  Lim  x )  ->  (
( A  +o  x
)  e.  On  <->  U_ y  e.  x  ( A  +o  y )  e.  On ) )
2319, 22syl5ibr 213 . . . 4  |-  ( ( A  e.  On  /\  Lim  x )  ->  ( A. y  e.  x  ( A  +o  y
)  e.  On  ->  ( A  +o  x )  e.  On ) )
2423expcom 425 . . 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 4811 . 2  |-  ( B  e.  On  ->  ( A  e.  On  ->  ( A  +o  B )  e.  On ) )
2625impcom 420 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   _Vcvv 2924   (/)c0 3596   U_ciun 4061   Oncon0 4549   Lim wlim 4550   suc csuc 4551  (class class class)co 6048    +o coa 6688
This theorem is referenced by:  omcl  6747  oaord  6757  oacan  6758  oaword  6759  oawordri  6760  oawordeulem  6764  oalimcl  6770  oaass  6771  oaf1o  6773  odi  6789  omopth2  6794  oeoalem  6806  oeoa  6807  oancom  7570  cantnfvalf  7584  dfac12lem2  7988  cdanum  8043  wunex3  8580
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-oadd 6695
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