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Theorem alephgch 9048
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 8449 . . . . 5  |-  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )
2 sdomen2 7659 . . . . . 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 1695 . . 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 5874 . . 3  |-  ( aleph `  A )  e.  _V
8 elgch 8996 . . 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 1377    e. wcel 1767   _Vcvv 3113   ~Pcpw 4010   class class class wbr 4447   suc csuc 4880   ` cfv 5586    ~~ cen 7510    ~< csdm 7512   Fincfn 7513   alephcale 8313  GCHcgch 8994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-om 6679  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-oi 7931  df-har 7980  df-card 8316  df-aleph 8317  df-gch 8995
This theorem is referenced by:  gch3  9050
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