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Theorem gch3 8555
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 448 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  x  e.  On )
2 fvex 5742 . . . . 5  |-  ( aleph `  x )  e.  _V
3 simpl 444 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  On )  -> GCH  =  _V )
42, 3syl5eleqr 2523 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
5 fvex 5742 . . . . 5  |-  ( aleph ` 
suc  x )  e. 
_V
65, 3syl5eleqr 2523 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
7 gchaleph2 8551 . . . 4  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
81, 4, 6, 7syl3anc 1184 . . 3  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
98ralrimiva 2789 . 2  |-  (GCH  =  _V  ->  A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
10 alephgch 8553 . . . . . 6  |-  ( (
aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ( aleph `  x )  e. GCH )
1110ralimi 2781 . . . . 5  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  A. x  e.  On  ( aleph `  x
)  e. GCH )
12 alephfnon 7946 . . . . . 6  |-  aleph  Fn  On
13 ffnfv 5894 . . . . . 6  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x )  e. GCH ) )
1412, 13mpbiran 885 . . . . 5  |-  ( aleph : On -->GCH 
<-> 
A. x  e.  On  ( aleph `  x )  e. GCH )
1511, 14sylibr 204 . . . 4  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  aleph : On -->GCH )
16 df-f 5458 . . . . 5  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  ran  aleph  C_ GCH ) )
1712, 16mpbiran 885 . . . 4  |-  ( aleph : On -->GCH 
<->  ran  aleph  C_ GCH )
1815, 17sylib 189 . . 3  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ran  aleph  C_ GCH )
19 gch2 8554 . . 3  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
2018, 19sylibr 204 . 2  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  -> GCH  =  _V )
219, 20impbii 181 1  |-  (GCH  =  _V 
<-> 
A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   class class class wbr 4212   Oncon0 4581   suc csuc 4583   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454    ~~ cen 7106   alephcale 7823  GCHcgch 8495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-reg 7560  ax-inf2 7596
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-seqom 6705  df-1o 6724  df-2o 6725  df-oadd 6728  df-omul 6729  df-oexp 6730  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-oi 7479  df-har 7526  df-wdom 7527  df-cnf 7617  df-r1 7690  df-rank 7691  df-card 7826  df-aleph 7827  df-ac 7997  df-cda 8048  df-fin4 8167  df-gch 8496
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