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Theorem alephexp2 8990
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8988 (which works if the base is less than or equal to the exponent) and infmap 8985 (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 8497 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5861 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7601 . . . . 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 197 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 domrefg 7590 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( aleph `  A
)  ~<_  ( aleph `  A
) )
72, 6ax-mp 5 . . 3  |-  ( aleph `  A )  ~<_  ( aleph `  A )
8 infmap 8985 . . 3  |-  ( ( om  ~<_  ( aleph `  A
)  /\  ( aleph `  A )  ~<_  ( aleph `  A ) )  -> 
( ( aleph `  A
)  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
95, 7, 8sylancl 662 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } )
10 pm3.2 447 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On )
) )
1110pm2.43i 48 . . . 4  |-  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On ) )
12 ssid 3463 . . . 4  |-  A  C_  A
13 alephexp1 8988 . . . 4  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  A  C_  A
)  ->  ( ( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
1411, 12, 13sylancl 662 . . 3  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
15 enen1 7697 . . 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 17 . 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 212 1  |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A
) )  ~~  {
x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    /\ wa 369    e. wcel 1844   {cab 2389   _Vcvv 3061    C_ wss 3416   class class class wbr 4397   Oncon0 5412   ` cfv 5571  (class class class)co 6280   omcom 6685   2oc2o 7163    ^m cmap 7459    ~~ cen 7553    ~<_ cdom 7554   alephcale 8351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-inf2 8093  ax-ac2 8877
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-tp 3979  df-op 3981  df-uni 4194  df-int 4230  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-tr 4492  df-eprel 4736  df-id 4740  df-po 4746  df-so 4747  df-fr 4784  df-se 4785  df-we 4786  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-pred 5369  df-ord 5415  df-on 5416  df-lim 5417  df-suc 5418  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-isom 5580  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-om 6686  df-1st 6786  df-2nd 6787  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-1o 7169  df-2o 7170  df-oadd 7173  df-er 7350  df-map 7461  df-en 7557  df-dom 7558  df-sdom 7559  df-fin 7560  df-oi 7971  df-har 8020  df-card 8354  df-aleph 8355  df-acn 8357  df-ac 8531
This theorem is referenced by:  gch-kn  9087
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