MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephexp1 Unicode version

Theorem alephexp1 8410
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 7906 . . . 4  |-  ( aleph `  B )  e.  On
2 onenon 7792 . . . 4  |-  ( (
aleph `  B )  e.  On  ->  ( aleph `  B )  e.  dom  card )
31, 2mp1i 12 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  B )  e.  dom  card )
4 fvex 5701 . . . 4  |-  ( aleph `  B )  e.  _V
5 simplr 732 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  B  e.  On )
6 alephgeom 7919 . . . . 5  |-  ( B  e.  On  <->  om  C_  ( aleph `  B ) )
75, 6sylib 189 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  B ) )
8 ssdomg 7112 . . . 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 5701 . . . 4  |-  ( aleph `  A )  e.  _V
11 ordom 4813 . . . . . 6  |-  Ord  om
12 2onn 6842 . . . . . 6  |-  2o  e.  om
13 ordelss 4557 . . . . . 6  |-  ( ( Ord  om  /\  2o  e.  om )  ->  2o  C_ 
om )
1411, 12, 13mp2an 654 . . . . 5  |-  2o  C_  om
15 simpll 731 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  A  e.  On )
16 alephgeom 7919 . . . . . 6  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
1715, 16sylib 189 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  om  C_  ( aleph `  A ) )
1814, 17syl5ss 3319 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  2o  C_  ( aleph `  A ) )
19 ssdomg 7112 . . . 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 7915 . . . . . 6  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  <->  (
aleph `  A )  C_  ( aleph `  B )
) )
22 ssdomg 7112 . . . . . . 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 220 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  B  ->  ( aleph `  A )  ~<_  ( aleph `  B )
) )
2524imp 419 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ( aleph `  B ) )
264canth2 7219 . . . . 5  |-  ( aleph `  B )  ~<  ~P ( aleph `  B )
27 sdomdom 7094 . . . . 5  |-  ( (
aleph `  B )  ~<  ~P ( aleph `  B )  ->  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)
2826, 27ax-mp 8 . . . 4  |-  ( aleph `  B )  ~<_  ~P ( aleph `  B )
29 domtr 7119 . . . 4  |-  ( ( ( aleph `  A )  ~<_  ( aleph `  B )  /\  ( aleph `  B )  ~<_  ~P ( aleph `  B )
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
3025, 28, 29sylancl 644 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  A  C_  B
)  ->  ( aleph `  A )  ~<_  ~P ( aleph `  B ) )
31 mappwen 7949 . . 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 7174 . . 3  |-  ~P ( aleph `  B )  ~~  ( 2o  ^m  ( aleph `  B ) )
34 enen2 7207 . . 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 189 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 177    /\ wa 359    e. wcel 1721   _Vcvv 2916    C_ wss 3280   ~Pcpw 3759   class class class wbr 4172   Ord word 4540   Oncon0 4541   omcom 4804   dom cdm 4837   ` cfv 5413  (class class class)co 6040   2oc2o 6677    ^m cmap 6977    ~~ cen 7065    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778   alephcale 7779
This theorem is referenced by:  alephexp2  8412
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783
  Copyright terms: Public domain W3C validator