MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  alephexp2 Unicode version

Theorem alephexp2 8390
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8388 (which works if the base is less than or equal to the exponent) and infmap 8385 (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 7897 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5683 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7090 . . . . 5  |-  ( (
aleph `  A )  e. 
_V  ->  ( om  C_  ( aleph `  A )  ->  om 
~<_  ( aleph `  A )
) )
42, 3ax-mp 8 . . . 4  |-  ( om  C_  ( aleph `  A )  ->  om  ~<_  ( aleph `  A
) )
51, 4sylbi 188 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 domrefg 7079 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( aleph `  A
)  ~<_  ( aleph `  A
) )
72, 6ax-mp 8 . . 3  |-  ( aleph `  A )  ~<_  ( aleph `  A )
8 infmap 8385 . . 3  |-  ( ( om  ~<_  ( aleph `  A
)  /\  ( aleph `  A )  ~<_  ( aleph `  A ) )  -> 
( ( aleph `  A
)  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
95, 7, 8sylancl 644 . 2  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } )
10 pm3.2 435 . . . . 5  |-  ( A  e.  On  ->  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On )
) )
1110pm2.43i 45 . . . 4  |-  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On ) )
12 ssid 3311 . . . 4  |-  A  C_  A
13 alephexp1 8388 . . . 4  |-  ( ( ( A  e.  On  /\  A  e.  On )  /\  A  C_  A
)  ->  ( ( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
1411, 12, 13sylancl 644 . . 3  |-  ( A  e.  On  ->  (
( aleph `  A )  ^m  ( aleph `  A )
)  ~~  ( 2o  ^m  ( aleph `  A )
) )
15 enen1 7184 . . 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 202 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 177    /\ wa 359    e. wcel 1717   {cab 2374   _Vcvv 2900    C_ wss 3264   class class class wbr 4154   Oncon0 4523   omcom 4786   ` cfv 5395  (class class class)co 6021   2oc2o 6655    ^m cmap 6955    ~~ cen 7043    ~<_ cdom 7044   alephcale 7757
This theorem is referenced by:  gch-kn  8490
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-ac2 8277
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-oi 7413  df-har 7460  df-card 7760  df-aleph 7761  df-acn 7763  df-ac 7931
  Copyright terms: Public domain W3C validator