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Theorem alephexp2 8487
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8485 (which works if the base is less than or equal to the exponent) and infmap 8482 (which works if the exponent is less than or equal to the base). They can be equated only when the base is equal to the exponent, and this is the result. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
alephexp2  |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A
) )  ~~  {
x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
Distinct variable group:    x, A

Proof of Theorem alephexp2
StepHypRef Expression
1 alephgeom 7994 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5771 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7182 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
42, 3ax-mp 5 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
51, 4sylbi 189 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 domrefg 7171 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( aleph `  A
)  ~<_  ( aleph `  A
) )
72, 6ax-mp 5 . . 3  |-  ( aleph `  A )  ~<_  ( aleph `  A )
8 infmap 8482 . . 3  |-  ( ( om  ~<_  ( aleph `  A
)  /\  ( aleph `  A )  ~<_  ( aleph `  A ) )  -> 
( ( aleph `  A
)  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
95, 7, 8sylancl 645 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } )
10 pm3.2 436 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On )
) )
1110pm2.43i 46 . . . 4  |-  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On ) )
12 ssid 3353 . . . 4  |-  A  C_  A
13 alephexp1 8485 . . . 4  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  A  C_  A
)  ->  ( ( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
1411, 12, 13sylancl 645 . . 3  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
15 enen1 7276 . . 3  |-  ( ( ( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
)  ->  ( (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) }  <-> 
( 2o  ^m  ( aleph `  A ) ) 
~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A
) ) } ) )
1614, 15syl 16 . 2  |-  ( A  e.  On  ->  (
( ( aleph `  A
)  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) }  <->  ( 2o  ^m  ( aleph `  A )
)  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } ) )
179, 16mpbid 203 1  |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A
) )  ~~  {
x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1727   {cab 2428   _Vcvv 2962    C_ wss 3306   class class class wbr 4237   Oncon0 4610   omcom 4874   ` cfv 5483  (class class class)co 6110   2oc2o 6747    ^m cmap 7047    ~~ cen 7135    ~<_ cdom 7136   alephcale 7854
This theorem is referenced by:  gch-kn  8583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-inf2 7625  ax-ac2 8374
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-se 4571  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-isom 5492  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-recs 6662  df-rdg 6697  df-1o 6753  df-2o 6754  df-oadd 6757  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-fin 7142  df-oi 7508  df-har 7555  df-card 7857  df-aleph 7858  df-acn 7860  df-ac 8028
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