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Theorem oecl 6552
Description: Closure law for ordinal exponentiation. (Contributed by NM, 1-Jan-2005.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
oecl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )

Proof of Theorem oecl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5882 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6535 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  =  1o
3 1on 6502 . . . . . . . . 9  |-  1o  e.  On
42, 3eqeltri 2366 . . . . . . . 8  |-  ( (/)  ^o  (/) )  e.  On
51, 4syl6eqel 2384 . . . . . . 7  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
65adantl 452 . . . . . 6  |-  ( ( B  e.  On  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
7 oe0m1 6536 . . . . . . . . 9  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
87biimpa 470 . . . . . . . 8  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
9 0elon 4461 . . . . . . . 8  |-  (/)  e.  On
108, 9syl6eqel 2384 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
1110adantll 694 . . . . . 6  |-  ( ( ( B  e.  On  /\  B  e.  On )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
126, 11oe0lem 6528 . . . . 5  |-  ( ( B  e.  On  /\  B  e.  On )  ->  ( (/)  ^o  B )  e.  On )
1312anidms 626 . . . 4  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
14 oveq1 5881 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
1514eleq1d 2362 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  e.  On  <->  ( (/)  ^o  B
)  e.  On ) )
1613, 15syl5ibr 212 . . 3  |-  ( A  =  (/)  ->  ( B  e.  On  ->  ( A  ^o  B )  e.  On ) )
1716impcom 419 . 2  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  e.  On )
18 oveq2 5882 . . . . . . 7  |-  ( x  =  (/)  ->  ( A  ^o  x )  =  ( A  ^o  (/) ) )
1918eleq1d 2362 . . . . . 6  |-  ( x  =  (/)  ->  ( ( A  ^o  x )  e.  On  <->  ( A  ^o  (/) )  e.  On ) )
20 oveq2 5882 . . . . . . 7  |-  ( x  =  y  ->  ( A  ^o  x )  =  ( A  ^o  y
) )
2120eleq1d 2362 . . . . . 6  |-  ( x  =  y  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  y )  e.  On ) )
22 oveq2 5882 . . . . . . 7  |-  ( x  =  suc  y  -> 
( A  ^o  x
)  =  ( A  ^o  suc  y ) )
2322eleq1d 2362 . . . . . 6  |-  ( x  =  suc  y  -> 
( ( A  ^o  x )  e.  On  <->  ( A  ^o  suc  y
)  e.  On ) )
24 oveq2 5882 . . . . . . 7  |-  ( x  =  B  ->  ( A  ^o  x )  =  ( A  ^o  B
) )
2524eleq1d 2362 . . . . . 6  |-  ( x  =  B  ->  (
( A  ^o  x
)  e.  On  <->  ( A  ^o  B )  e.  On ) )
26 oe0 6537 . . . . . . . 8  |-  ( A  e.  On  ->  ( A  ^o  (/) )  =  1o )
2726, 3syl6eqel 2384 . . . . . . 7  |-  ( A  e.  On  ->  ( A  ^o  (/) )  e.  On )
2827adantr 451 . . . . . 6  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( A  ^o  (/) )  e.  On )
29 omcl 6551 . . . . . . . . . . 11  |-  ( ( ( A  ^o  y
)  e.  On  /\  A  e.  On )  ->  ( ( A  ^o  y )  .o  A
)  e.  On )
3029expcom 424 . . . . . . . . . 10  |-  ( A  e.  On  ->  (
( A  ^o  y
)  e.  On  ->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3130adantr 451 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( ( A  ^o  y )  .o  A
)  e.  On ) )
32 oesuc 6542 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( A  ^o  suc  y )  =  ( ( A  ^o  y
)  .o  A ) )
3332eleq1d 2362 . . . . . . . . 9  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  suc  y )  e.  On  <->  ( ( A  ^o  y
)  .o  A )  e.  On ) )
3431, 33sylibrd 225 . . . . . . . 8  |-  ( ( A  e.  On  /\  y  e.  On )  ->  ( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) )
3534expcom 424 . . . . . . 7  |-  ( y  e.  On  ->  ( A  e.  On  ->  ( ( A  ^o  y
)  e.  On  ->  ( A  ^o  suc  y
)  e.  On ) ) )
3635adantrd 454 . . . . . 6  |-  ( y  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( ( A  ^o  y )  e.  On  ->  ( A  ^o  suc  y )  e.  On ) ) )
37 vex 2804 . . . . . . . . 9  |-  x  e. 
_V
38 iunon 6371 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  A. y  e.  x  ( A  ^o  y )  e.  On )  ->  U_ y  e.  x  ( A  ^o  y
)  e.  On )
3937, 38mpan 651 . . . . . . . 8  |-  ( A. y  e.  x  ( A  ^o  y )  e.  On  ->  U_ y  e.  x  ( A  ^o  y )  e.  On )
40 oelim 6549 . . . . . . . . . . . 12  |-  ( ( ( A  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  A )  ->  ( A  ^o  x )  =  U_ y  e.  x  ( A  ^o  y ) )
4137, 40mpanlr1 667 . . . . . . . . . . 11  |-  ( ( ( A  e.  On  /\ 
Lim  x )  /\  (/) 
e.  A )  -> 
( A  ^o  x
)  =  U_ y  e.  x  ( A  ^o  y ) )
4241anasss 628 . . . . . . . . . 10  |-  ( ( A  e.  On  /\  ( Lim  x  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4342an12s 776 . . . . . . . . 9  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A  ^o  x )  = 
U_ y  e.  x  ( A  ^o  y
) )
4443eleq1d 2362 . . . . . . . 8  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  (
( A  ^o  x
)  e.  On  <->  U_ y  e.  x  ( A  ^o  y )  e.  On ) )
4539, 44syl5ibr 212 . . . . . . 7  |-  ( ( Lim  x  /\  ( A  e.  On  /\  (/)  e.  A
) )  ->  ( A. y  e.  x  ( A  ^o  y
)  e.  On  ->  ( A  ^o  x )  e.  On ) )
4645ex 423 . . . . . 6  |-  ( Lim  x  ->  ( ( A  e.  On  /\  (/)  e.  A
)  ->  ( A. y  e.  x  ( A  ^o  y )  e.  On  ->  ( A  ^o  x )  e.  On ) ) )
4719, 21, 23, 25, 28, 36, 46tfinds3 4671 . . . . 5  |-  ( B  e.  On  ->  (
( A  e.  On  /\  (/)  e.  A )  -> 
( A  ^o  B
)  e.  On ) )
4847exp3a 425 . . . 4  |-  ( B  e.  On  ->  ( A  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
4948com12 27 . . 3  |-  ( A  e.  On  ->  ( B  e.  On  ->  (
(/)  e.  A  ->  ( A  ^o  B )  e.  On ) ) )
5049imp31 421 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  e.  On )
5117, 50oe0lem 6528 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468   U_ciun 3921   Oncon0 4408   Lim wlim 4409   suc csuc 4410  (class class class)co 5874   1oc1o 6488    .o comu 6493    ^o coe 6494
This theorem is referenced by:  oen0  6600  oeordi  6601  oeord  6602  oecan  6603  oeword  6604  oewordri  6606  oeworde  6607  oeordsuc  6608  oeoalem  6610  oeoa  6611  oeoelem  6612  oeoe  6613  oelimcl  6614  oeeulem  6615  oeeui  6616  oaabs2  6659  omabs  6661  cantnfle  7388  cantnflt  7389  cantnfp1  7399  cantnflem1d  7406  cantnflem1  7407  cantnflem2  7408  cantnflem3  7409  cantnflem4  7410  cantnf  7411  oemapwe  7412  cantnffval2  7413  cnfcomlem  7418  cnfcom  7419  cnfcom3lem  7422  cnfcom3  7423  infxpenc  7661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-omul 6500  df-oexp 6501
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