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Theorem oasuc 6160
Description: Addition with successor. Definition 8.1 of [TakeutiZaring] p. 56. (Contributed by NM, 3-May-1995.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oasuc  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )

Proof of Theorem oasuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 suceloni 4284 . . . . . 6  |-  ( B  e.  On  ->  suc  B  e.  On )
2 oav2 6159 . . . . . 6  |-  ( ( A  e.  On  /\  suc  B  e.  On )  ->  ( A  +o  suc  B )  =  ( A  u.  U_ x  e.  suc  B  suc  ( A  +o  x ) ) )
31, 2sylan2 280 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  ( A  u.  U_ x  e. 
suc  B  suc  ( A  +o  x ) ) )
4 df-suc 4165 . . . . . . . . . 10  |-  suc  B  =  ( B  u.  { B } )
5 iuneq1 3720 . . . . . . . . . 10  |-  ( suc 
B  =  ( B  u.  { B }
)  ->  U_ x  e. 
suc  B  suc  ( A  +o  x )  = 
U_ x  e.  ( B  u.  { B } ) suc  ( A  +o  x ) )
64, 5ax-mp 7 . . . . . . . . 9  |-  U_ x  e.  suc  B  suc  ( A  +o  x )  = 
U_ x  e.  ( B  u.  { B } ) suc  ( A  +o  x )
7 iunxun 3785 . . . . . . . . 9  |-  U_ x  e.  ( B  u.  { B } ) suc  ( A  +o  x )  =  ( U_ x  e.  B  suc  ( A  +o  x )  u. 
U_ x  e.  { B } suc  ( A  +o  x ) )
86, 7eqtri 2105 . . . . . . . 8  |-  U_ x  e.  suc  B  suc  ( A  +o  x )  =  ( U_ x  e.  B  suc  ( A  +o  x )  u. 
U_ x  e.  { B } suc  ( A  +o  x ) )
9 oveq2 5602 . . . . . . . . . . 11  |-  ( x  =  B  ->  ( A  +o  x )  =  ( A  +o  B
) )
10 suceq 4196 . . . . . . . . . . 11  |-  ( ( A  +o  x )  =  ( A  +o  B )  ->  suc  ( A  +o  x
)  =  suc  ( A  +o  B ) )
119, 10syl 14 . . . . . . . . . 10  |-  ( x  =  B  ->  suc  ( A  +o  x
)  =  suc  ( A  +o  B ) )
1211iunxsng 3782 . . . . . . . . 9  |-  ( B  e.  On  ->  U_ x  e.  { B } suc  ( A  +o  x
)  =  suc  ( A  +o  B ) )
1312uneq2d 3140 . . . . . . . 8  |-  ( B  e.  On  ->  ( U_ x  e.  B  suc  ( A  +o  x
)  u.  U_ x  e.  { B } suc  ( A  +o  x
) )  =  (
U_ x  e.  B  suc  ( A  +o  x
)  u.  suc  ( A  +o  B ) ) )
148, 13syl5eq 2129 . . . . . . 7  |-  ( B  e.  On  ->  U_ x  e.  suc  B  suc  ( A  +o  x )  =  ( U_ x  e.  B  suc  ( A  +o  x )  u. 
suc  ( A  +o  B ) ) )
1514uneq2d 3140 . . . . . 6  |-  ( B  e.  On  ->  ( A  u.  U_ x  e. 
suc  B  suc  ( A  +o  x ) )  =  ( A  u.  ( U_ x  e.  B  suc  ( A  +o  x
)  u.  suc  ( A  +o  B ) ) ) )
1615adantl 271 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  u.  U_ x  e.  suc  B  suc  ( A  +o  x
) )  =  ( A  u.  ( U_ x  e.  B  suc  ( A  +o  x
)  u.  suc  ( A  +o  B ) ) ) )
173, 16eqtrd 2117 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  ( A  u.  ( U_ x  e.  B  suc  ( A  +o  x )  u. 
suc  ( A  +o  B ) ) ) )
18 unass 3143 . . . 4  |-  ( ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) )  u.  suc  ( A  +o  B ) )  =  ( A  u.  ( U_ x  e.  B  suc  ( A  +o  x
)  u.  suc  ( A  +o  B ) ) )
1917, 18syl6eqr 2135 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  ( ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) )  u.  suc  ( A  +o  B ) ) )
20 oav2 6159 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  B
)  =  ( A  u.  U_ x  e.  B  suc  ( A  +o  x ) ) )
2120uneq1d 3139 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  u.  suc  ( A  +o  B
) )  =  ( ( A  u.  U_ x  e.  B  suc  ( A  +o  x
) )  u.  suc  ( A  +o  B
) ) )
2219, 21eqtr4d 2120 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  ( ( A  +o  B )  u.  suc  ( A  +o  B ) ) )
23 sssucid 4209 . . 3  |-  ( A  +o  B )  C_  suc  ( A  +o  B
)
24 ssequn1 3156 . . 3  |-  ( ( A  +o  B ) 
C_  suc  ( A  +o  B )  <->  ( ( A  +o  B )  u. 
suc  ( A  +o  B ) )  =  suc  ( A  +o  B ) )
2523, 24mpbi 143 . 2  |-  ( ( A  +o  B )  u.  suc  ( A  +o  B ) )  =  suc  ( A  +o  B )
2622, 25syl6eq 2133 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  +o  suc  B )  =  suc  ( A  +o  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1287    e. wcel 1436    u. cun 2984    C_ wss 2986   {csn 3425   U_ciun 3707   Oncon0 4157   suc csuc 4159  (class class class)co 5594    +o coa 6113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-suc 4165  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-irdg 6070  df-oadd 6120
This theorem is referenced by:  onasuc  6162  nnaordi  6200
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