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Theorem alephexp1 8081
Description: An exponentiation law for alephs. Lemma 6.1 of [Jech] p. 42. (Contributed by NM, 29-Sep-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
alephexp1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )

Proof of Theorem alephexp1
StepHypRef Expression
1 alephon 7580 . . . 4  |-  ( aleph `  B )  e.  On
2 onenon 7466 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
31, 2mp1i 13 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  B )  e.  dom  card )
4 fvex 5391 . . . 4  |-  ( aleph `  B )  e.  _V
5 simplr 734 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  B  e.  On )
6 alephgeom 7593 . . . . 5  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
75, 6sylib 190 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  B ) )
8 ssdomg 6793 . . . 4  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
94, 7, 8mpsyl 61 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  ~<_  ( aleph `  B ) )
10 fvex 5391 . . . 4  |-  ( aleph `  A )  e.  _V
11 ordom 4556 . . . . . 6  |-  Ord  om
12 2onn 6524 . . . . . 6  |-  2o  e.  om
13 ordelss 4301 . . . . . 6  |-  ( ( Ord  om  /\  2o  e.  om )  ->  2o  C_ 
om )
1411, 12, 13mp2an 656 . . . . 5  |-  2o  C_  om
15 simpll 733 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  A  e.  On )
16 alephgeom 7593 . . . . . 6  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1715, 16sylib 190 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  A ) )
1814, 17syl5ss 3111 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  C_  ( aleph `  A ) )
19 ssdomg 6793 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( 2o  C_  ( aleph `  A )  ->  2o  ~<_  ( aleph `  A
) ) )
2010, 18, 19mpsyl 61 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  ~<_  ( aleph `  A ) )
21 alephord3 7589 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
22 ssdomg 6793 . . . . . . 7  |-  ( (
aleph `  B )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  B )  -> 
( aleph `  A )  ~<_  ( aleph `  B )
) )
234, 22ax-mp 10 . . . . . 6  |-  ( (
aleph `  A )  C_  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
2421, 23syl6bi 221 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
2524imp 420 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ( aleph `  B ) )
264canth2 6899 . . . . 5  |-  ( aleph `  B )  ~<  ~P ( aleph `  B )
27 sdomdom 6775 . . . . 5  |-  ( (
aleph `  B )  ~<  ~P ( aleph `  B )  ->  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)
2826, 27ax-mp 10 . . . 4  |-  ( aleph `  B )  ~<_  ~P ( aleph `  B )
29 domtr 6799 . . . 4  |-  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
3025, 28, 29sylancl 646 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
31 mappwen 7623 . . 3  |-  ( ( ( ( aleph `  B
)  e.  dom  card  /\ 
om  ~<_  ( aleph `  B
) )  /\  ( 2o 
~<_  ( aleph `  A )  /\  ( aleph `  A )  ~<_  ~P ( aleph `  B )
) )  ->  (
( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
323, 9, 20, 30, 31syl22anc 1188 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
334pw2en 6854 . . 3  |-  ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )
34 enen2 6887 . . 3  |-  ( ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )  ->  ( ( (
aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B )  <->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) ) )
3533, 34ax-mp 10 . 2  |-  ( ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B )  <->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )
3632, 35sylib 190 1  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1621   _Vcvv 2727    C_ wss 3078   ~Pcpw 3530   class class class wbr 3920   Ord word 4284   Oncon0 4285   omcom 4547   dom cdm 4580   ` cfv 4592  (class class class)co 5710   2oc2o 6359    ^m cmap 6658    ~~ cen 6746    ~<_ cdom 6747    ~< csdm 6748   cardccrd 7452   alephcale 7453
This theorem is referenced by:  alephexp2  8083
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-oi 7109  df-har 7156  df-card 7456  df-aleph 7457
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