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Theorem alephexp1 8197
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 7692 . . . 4  |-  ( aleph `  B )  e.  On
2 onenon 7578 . . . 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 5500 . . . 4  |-  ( aleph `  B )  e.  _V
5 simplr 733 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  B  e.  On )
6 alephgeom 7705 . . . . 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 6903 . . . 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 5500 . . . 4  |-  ( aleph `  A )  e.  _V
11 ordom 4665 . . . . . 6  |-  Ord  om
12 2onn 6634 . . . . . 6  |-  2o  e.  om
13 ordelss 4408 . . . . . 6  |-  ( ( Ord  om  /\  2o  e.  om )  ->  2o  C_ 
om )
1411, 12, 13mp2an 655 . . . . 5  |-  2o  C_  om
15 simpll 732 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  A  e.  On )
16 alephgeom 7705 . . . . . 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 3192 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  C_  ( aleph `  A ) )
19 ssdomg 6903 . . . 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 7701 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
22 ssdomg 6903 . . . . . . 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 7010 . . . . 5  |-  ( aleph `  B )  ~<  ~P ( aleph `  B )
27 sdomdom 6885 . . . . 5  |-  ( (
aleph `  B )  ~<  ~P ( aleph `  B )  ->  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)
2826, 27ax-mp 10 . . . 4  |-  ( aleph `  B )  ~<_  ~P ( aleph `  B )
29 domtr 6910 . . . 4  |-  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
3025, 28, 29sylancl 645 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
31 mappwen 7735 . . 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 1185 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
334pw2en 6965 . . 3  |-  ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )
34 enen2 6998 . . 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 1685   _Vcvv 2790    C_ wss 3154   ~Pcpw 3627   class class class wbr 4025   Ord word 4391   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5222  (class class class)co 5820   2oc2o 6469    ^m cmap 6768    ~~ cen 6856    ~<_ cdom 6857    ~< csdm 6858   cardccrd 7564   alephcale 7565
This theorem is referenced by:  alephexp2  8199
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-2o 6476  df-oadd 6479  df-er 6656  df-map 6770  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-oi 7221  df-har 7268  df-card 7568  df-aleph 7569
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