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Theorem oacl 6455
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 6449 . 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 2740 . . . . . . . 8  |-  w  e. 
_V
4 suceq 4399 . . . . . . . . 9  |-  ( z  =  w  ->  suc  z  =  suc  w )
5 eqid 2177 . . . . . . . . 9  |-  ( z  e.  _V  |->  suc  z
)  =  ( z  e.  _V  |->  suc  z
)
63sucex 4495 . . . . . . . . 9  |-  suc  w  e.  _V
74, 5, 6fvmpt 5589 . . . . . . . 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 2243 . . . . . 6  |-  ( ( ( z  e.  _V  |->  suc  z ) `  w
)  e.  On  <->  suc  w  e.  On )
109ralbii 2483 . . . . 5  |-  ( A. w  e.  On  (
( z  e.  _V  |->  suc  z ) `  w
)  e.  On  <->  A. w  e.  On  suc  w  e.  On )
11 onsuc 4497 . . . . 5  |-  ( w  e.  On  ->  suc  w  e.  On )
1210, 11mprgbir 2535 . . . 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 6381 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( rec ( ( z  e.  _V  |->  suc  z ) ,  A
) `  B )  e.  On )
151, 14eqeltrd 2254 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 1353    e. wcel 2148   A.wral 2455   _Vcvv 2737    |-> cmpt 4061   Oncon0 4360   suc csuc 4362   ` cfv 5212  (class class class)co 5869   reccrdg 6364    +o coa 6408
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-recs 6300  df-irdg 6365  df-oadd 6415
This theorem is referenced by:  omcl  6456  omv2  6460
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