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

Theorem alephgch 9069
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 8470 . . . . 5  |-  -.  (
( aleph `  A )  ~<  x  /\  x  ~<  (
aleph `  suc  A ) )
2 sdomen2 7681 . . . . . 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 1720 . . 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 5882 . . 3  |-  ( aleph `  A )  e.  _V
8 elgch 9017 . . 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 1393    e. wcel 1819   _Vcvv 3109   ~Pcpw 4015   class class class wbr 4456   suc csuc 4889   ` cfv 5594    ~~ cen 7532    ~< csdm 7534   Fincfn 7535   alephcale 8334  GCHcgch 9015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-om 6700  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-har 8002  df-card 8337  df-aleph 8338  df-gch 9016
This theorem is referenced by:  gch3  9071
  Copyright terms: Public domain W3C validator