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Theorem alephexp1 8201
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 7696 . . . 4  |-  ( aleph `  B )  e.  On
2 onenon 7582 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
31, 2mp1i 11 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  B )  e.  dom  card )
4 fvex 5539 . . . 4  |-  ( aleph `  B )  e.  _V
5 simplr 731 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  B  e.  On )
6 alephgeom 7709 . . . . 5  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
75, 6sylib 188 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  B ) )
8 ssdomg 6907 . . . 4  |-  ( (
aleph `  B )  e. 
_V  ->  ( om  C_  ( aleph `  B )  ->  om 
~<_  ( aleph `  B )
) )
94, 7, 8mpsyl 59 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  ~<_  ( aleph `  B ) )
10 fvex 5539 . . . 4  |-  ( aleph `  A )  e.  _V
11 ordom 4665 . . . . . 6  |-  Ord  om
12 2onn 6638 . . . . . 6  |-  2o  e.  om
13 ordelss 4408 . . . . . 6  |-  ( ( Ord  om  /\  2o  e.  om )  ->  2o  C_ 
om )
1411, 12, 13mp2an 653 . . . . 5  |-  2o  C_  om
15 simpll 730 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  A  e.  On )
16 alephgeom 7709 . . . . . 6  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1715, 16sylib 188 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  A ) )
1814, 17syl5ss 3190 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  C_  ( aleph `  A ) )
19 ssdomg 6907 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( 2o  C_  ( aleph `  A )  ->  2o  ~<_  ( aleph `  A
) ) )
2010, 18, 19mpsyl 59 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  ~<_  ( aleph `  A ) )
21 alephord3 7705 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
22 ssdomg 6907 . . . . . . 7  |-  ( (
aleph `  B )  e. 
_V  ->  ( ( aleph `  A )  C_  ( aleph `  B )  -> 
( aleph `  A )  ~<_  ( aleph `  B )
) )
234, 22ax-mp 8 . . . . . 6  |-  ( (
aleph `  A )  C_  ( aleph `  B )  ->  ( aleph `  A )  ~<_  ( aleph `  B )
)
2421, 23syl6bi 219 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
2524imp 418 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ( aleph `  B ) )
264canth2 7014 . . . . 5  |-  ( aleph `  B )  ~<  ~P ( aleph `  B )
27 sdomdom 6889 . . . . 5  |-  ( (
aleph `  B )  ~<  ~P ( aleph `  B )  ->  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)
2826, 27ax-mp 8 . . . 4  |-  ( aleph `  B )  ~<_  ~P ( aleph `  B )
29 domtr 6914 . . . 4  |-  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
3025, 28, 29sylancl 643 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
31 mappwen 7739 . . 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 1183 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B ) )
334pw2en 6969 . . 3  |-  ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )
34 enen2 7002 . . 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 8 . 2  |-  ( ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ~P ( aleph `  B )  <->  ( ( aleph `  A )  ^m  ( aleph `  B )
)  ~~  ( 2o  ^m  ( aleph `  B )
) )
3632, 35sylib 188 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 4    <-> wb 176    /\ wa 358    e. wcel 1684   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   class class class wbr 4023   Ord word 4391   Oncon0 4392   omcom 4656   dom cdm 4689   ` cfv 5255  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~~ cen 6860    ~<_ cdom 6861    ~< csdm 6862   cardccrd 7568   alephcale 7569
This theorem is referenced by:  alephexp2  8203
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  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-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573
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