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Theorem gch3 8297
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 5499 . . . . 5  |-  ( aleph `  x )  e.  _V
3 simpl 445 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  On )  -> GCH  =  _V )
42, 3syl5eleqr 2371 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
5 fvex 5499 . . . . 5  |-  ( aleph ` 
suc  x )  e. 
_V
65, 3syl5eleqr 2371 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
7 gchaleph2 8293 . . . 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 2627 . 2  |-  (GCH  =  _V  ->  A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
10 alephgch 8295 . . . . . 6  |-  ( (
aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ( aleph `  x )  e. GCH )
1110ralimi 2619 . . . . 5  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  A. x  e.  On  ( aleph `  x
)  e. GCH )
12 alephfnon 7687 . . . . . 6  |-  aleph  Fn  On
13 ffnfv 5646 . . . . . 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 5225 . . . . 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 8296 . . 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 1628    e. wcel 1688   A.wral 2544   _Vcvv 2789    C_ wss 3153   ~Pcpw 3626   class class class wbr 4024   Oncon0 4391   suc csuc 4393   ran crn 4689    Fn wfn 5216   -->wf 5217   ` cfv 5221    ~~ cen 6855   alephcale 7564  GCHcgch 8237
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-reg 7301  ax-inf2 7337
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-se 4352  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-isom 5230  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-seqom 6455  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-oexp 6480  df-er 6655  df-map 6769  df-en 6859  df-dom 6860  df-sdom 6861  df-fin 6862  df-oi 7220  df-har 7267  df-wdom 7268  df-cnf 7358  df-r1 7431  df-rank 7432  df-card 7567  df-aleph 7568  df-ac 7738  df-cda 7789  df-fin4 7908  df-gch 8238
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