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Theorem alephgch 8953
Description: If  ( aleph `  suc  A ) is equinumerous to the powerset of  ( aleph `  A
), then  ( aleph `  A
) is a GCH-set. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
alephgch  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( aleph `  A )  e. GCH )

Proof of Theorem alephgch
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 alephnbtwn2 8354 . . . . 5  |-  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )
2 sdomen2 7567 . . . . . 6  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( x  ~<  ( aleph `  suc  A )  <-> 
x  ~<  ~P ( aleph `  A ) ) )
32anbi2d 703 . . . . 5  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )  <->  ( ( aleph `  A )  ~<  x  /\  x  ~<  ~P ( aleph `  A ) ) ) )
41, 3mtbii 302 . . . 4  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  ~P ( aleph `  A )
) )
54alrimiv 1686 . . 3  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) )
65olcd 393 . 2  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( ( aleph `  A )  e. 
Fin  \/  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) ) )
7 fvex 5810 . . 3  |-  ( aleph `  A )  e.  _V
8 elgch 8901 . . 3  |-  ( (
aleph `  A )  e. 
_V  ->  ( ( aleph `  A )  e. GCH  <->  ( ( aleph `  A )  e. 
Fin  \/  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) ) ) )
97, 8ax-mp 5 . 2  |-  ( (
aleph `  A )  e. GCH  <->  ( ( aleph `  A )  e.  Fin  \/  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) ) )
106, 9sylibr 212 1  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( aleph `  A )  e. GCH )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369   A.wal 1368    e. wcel 1758   _Vcvv 3078   ~Pcpw 3969   class class class wbr 4401   suc csuc 4830   ` cfv 5527    ~~ cen 7418    ~< csdm 7420   Fincfn 7421   alephcale 8218  GCHcgch 8899
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
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-om 6588  df-recs 6943  df-rdg 6977  df-er 7212  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-gch 8900
This theorem is referenced by:  gch3  8955
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