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Theorem oacl 6486
Description: Closure law for ordinal addition. Proposition 8.2 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.) (Constructive proof by Jim Kingdon, 26-Jul-2019.)
Assertion
Ref Expression
oacl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )

Proof of Theorem oacl
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oav 6480 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( rec ( ( z  e. 
_V  |->  suc  z ) ,  A ) `  B
) )
2 id 19 . . 3  |-  ( A  e.  On  ->  A  e.  On )
3 vex 2755 . . . . . . . 8  |-  w  e. 
_V
4 suceq 4420 . . . . . . . . 9  |-  ( z  =  w  ->  suc  z  =  suc  w )
5 eqid 2189 . . . . . . . . 9  |-  ( z  e.  _V  |->  suc  z
)  =  ( z  e.  _V  |->  suc  z
)
63sucex 4516 . . . . . . . . 9  |-  suc  w  e.  _V
74, 5, 6fvmpt 5614 . . . . . . . 8  |-  ( w  e.  _V  ->  (
( z  e.  _V  |->  suc  z ) `  w
)  =  suc  w
)
83, 7ax-mp 5 . . . . . . 7  |-  ( ( z  e.  _V  |->  suc  z ) `  w
)  =  suc  w
98eleq1i 2255 . . . . . 6  |-  ( ( ( z  e.  _V  |->  suc  z ) `  w
)  e.  On  <->  suc  w  e.  On )
109ralbii 2496 . . . . 5  |-  ( A. w  e.  On  (
( z  e.  _V  |->  suc  z ) `  w
)  e.  On  <->  A. w  e.  On  suc  w  e.  On )
11 onsuc 4518 . . . . 5  |-  ( w  e.  On  ->  suc  w  e.  On )
1210, 11mprgbir 2548 . . . 4  |-  A. w  e.  On  ( ( z  e.  _V  |->  suc  z
) `  w )  e.  On
1312a1i 9 . . 3  |-  ( A  e.  On  ->  A. w  e.  On  ( ( z  e.  _V  |->  suc  z
) `  w )  e.  On )
142, 13rdgon 6412 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( z  e.  _V  |->  suc  z ) ,  A
) `  B )  e.  On )
151, 14eqeltrd 2266 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   _Vcvv 2752    |-> cmpt 4079   Oncon0 4381   suc csuc 4383   ` cfv 5235  (class class class)co 5897   reccrdg 6395    +o coa 6439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-recs 6331  df-irdg 6396  df-oadd 6446
This theorem is referenced by:  omcl  6487  omv2  6491
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