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Theorem alephgch 9098
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 8501 . . . . 5  |-  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )
2 sdomen2 7723 . . . . . 6  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( x  ~<  ( aleph `  suc  A )  <-> 
x  ~<  ~P ( aleph `  A ) ) )
32anbi2d 708 . . . . 5  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )  <->  ( ( aleph `  A )  ~<  x  /\  x  ~<  ~P ( aleph `  A ) ) ) )
41, 3mtbii 303 . . . 4  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  ~P ( aleph `  A )
) )
54alrimiv 1766 . . 3  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) )
65olcd 394 . 2  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( ( aleph `  A )  e. 
Fin  \/  A. x  -.  ( ( aleph `  A
)  ~<  x  /\  x  ~<  ~P ( aleph `  A
) ) ) )
7 fvex 5891 . . 3  |-  ( aleph `  A )  e.  _V
8 elgch 9046 . . 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 215 1  |-  ( (
aleph `  suc  A ) 
~~  ~P ( aleph `  A
)  ->  ( aleph `  A )  e. GCH )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370   A.wal 1435    e. wcel 1870   _Vcvv 3087   ~Pcpw 3985   class class class wbr 4426   suc csuc 5444   ` cfv 5601    ~~ cen 7574    ~< csdm 7576   Fincfn 7577   alephcale 8369  GCHcgch 9044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-om 6707  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-oi 8025  df-har 8073  df-card 8372  df-aleph 8373  df-gch 9045
This theorem is referenced by:  gch3  9100
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