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Theorem gch3 8256
Description: An equivalent formulation of the generalized continuum hypothesis. (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gch3  |-  (GCH  =  _V 
<-> 
A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )

Proof of Theorem gch3
StepHypRef Expression
1 simpr 449 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  x  e.  On )
2 fvex 5458 . . . . 5  |-  ( aleph `  x )  e.  _V
3 simpl 445 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  On )  -> GCH  =  _V )
42, 3syl5eleqr 2343 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
5 fvex 5458 . . . . 5  |-  ( aleph ` 
suc  x )  e. 
_V
65, 3syl5eleqr 2343 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
7 gchaleph2 8252 . . . 4  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
81, 4, 6, 7syl3anc 1187 . . 3  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
98ralrimiva 2599 . 2  |-  (GCH  =  _V  ->  A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
10 alephgch 8254 . . . . . 6  |-  ( (
aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ( aleph `  x )  e. GCH )
1110ralimi 2591 . . . . 5  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  A. x  e.  On  ( aleph `  x
)  e. GCH )
12 alephfnon 7646 . . . . . 6  |-  aleph  Fn  On
13 ffnfv 5605 . . . . . 6  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x )  e. GCH ) )
1412, 13mpbiran 889 . . . . 5  |-  ( aleph : On -->GCH 
<-> 
A. x  e.  On  ( aleph `  x )  e. GCH )
1511, 14sylibr 205 . . . 4  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  aleph : On -->GCH )
16 df-f 4671 . . . . 5  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  ran  aleph  C_ GCH ) )
1712, 16mpbiran 889 . . . 4  |-  ( aleph : On -->GCH 
<->  ran  aleph  C_ GCH )
1815, 17sylib 190 . . 3  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ran  aleph  C_ GCH )
19 gch2 8255 . . 3  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
2018, 19sylibr 205 . 2  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  -> GCH  =  _V )
219, 20impbii 182 1  |-  (GCH  =  _V 
<-> 
A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2757    C_ wss 3113   ~Pcpw 3585   class class class wbr 3983   Oncon0 4350   suc csuc 4352   ran crn 4648    Fn wfn 4654   -->wf 4655   ` cfv 4659    ~~ cen 6814   alephcale 7523  GCHcgch 8196
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-reg 7260  ax-inf2 7296
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-fal 1316  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-seqom 6414  df-1o 6433  df-2o 6434  df-oadd 6437  df-omul 6438  df-oexp 6439  df-er 6614  df-map 6728  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-oi 7179  df-har 7226  df-wdom 7227  df-cnf 7317  df-r1 7390  df-rank 7391  df-card 7526  df-aleph 7527  df-ac 7697  df-cda 7748  df-fin4 7867  df-gch 8197
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