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

Theorem alephexp2 8857
Description: An expression equinumerous to 2 to an aleph power. The proof equates the two laws for cardinal exponentiation alephexp1 8855 (which works if the base is less than or equal to the exponent) and infmap 8852 (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 8364 . . . 4  |-  ( A  e.  On  <->  om  C_  ( aleph `  A ) )
2 fvex 5810 . . . . 5  |-  ( aleph `  A )  e.  _V
3 ssdomg 7466 . . . . 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 195 . . 3  |-  ( A  e.  On  ->  om  ~<_  ( aleph `  A ) )
6 domrefg 7455 . . . 4  |-  ( (
aleph `  A )  e. 
_V  ->  ( aleph `  A
)  ~<_  ( aleph `  A
) )
72, 6ax-mp 5 . . 3  |-  ( aleph `  A )  ~<_  ( aleph `  A )
8 infmap 8852 . . 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 47 . . . 4  |-  ( A  e.  On  ->  ( A  e.  On  /\  A  e.  On ) )
12 ssid 3484 . . . 4  |-  A  C_  A
13 alephexp1 8855 . . . 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 7562 . . 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 210 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 184    /\ wa 369    e. wcel 1758   {cab 2439   _Vcvv 3078    C_ wss 3437   class class class wbr 4401   Oncon0 4828   ` cfv 5527  (class class class)co 6201   omcom 6587   2oc2o 7025    ^m cmap 7325    ~~ cen 7418    ~<_ cdom 7419   alephcale 8218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-ac2 8744
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-2o 7032  df-oadd 7035  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-oi 7836  df-har 7885  df-card 8221  df-aleph 8222  df-acn 8224  df-ac 8398
This theorem is referenced by:  gch-kn  8956
  Copyright terms: Public domain W3C validator