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Theorem gch3 8318
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 447 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  x  e.  On )
2 fvex 5555 . . . . 5  |-  ( aleph `  x )  e.  _V
3 simpl 443 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  On )  -> GCH  =  _V )
42, 3syl5eleqr 2383 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
5 fvex 5555 . . . . 5  |-  ( aleph ` 
suc  x )  e. 
_V
65, 3syl5eleqr 2383 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
7 gchaleph2 8314 . . . 4  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
81, 4, 6, 7syl3anc 1182 . . 3  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
98ralrimiva 2639 . 2  |-  (GCH  =  _V  ->  A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
10 alephgch 8316 . . . . . 6  |-  ( (
aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ( aleph `  x )  e. GCH )
1110ralimi 2631 . . . . 5  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  A. x  e.  On  ( aleph `  x
)  e. GCH )
12 alephfnon 7708 . . . . . 6  |-  aleph  Fn  On
13 ffnfv 5701 . . . . . 6  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x )  e. GCH ) )
1412, 13mpbiran 884 . . . . 5  |-  ( aleph : On -->GCH 
<-> 
A. x  e.  On  ( aleph `  x )  e. GCH )
1511, 14sylibr 203 . . . 4  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  aleph : On -->GCH )
16 df-f 5275 . . . . 5  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  ran  aleph  C_ GCH ) )
1712, 16mpbiran 884 . . . 4  |-  ( aleph : On -->GCH 
<->  ran  aleph  C_ GCH )
1815, 17sylib 188 . . 3  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ran  aleph  C_ GCH )
19 gch2 8317 . . 3  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
2018, 19sylibr 203 . 2  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  -> GCH  =  _V )
219, 20impbii 180 1  |-  (GCH  =  _V 
<-> 
A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   class class class wbr 4039   Oncon0 4408   suc csuc 4410   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271    ~~ cen 6876   alephcale 7585  GCHcgch 8258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-reg 7322  ax-inf2 7358
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-2o 6496  df-oadd 6499  df-omul 6500  df-oexp 6501  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-oi 7241  df-har 7288  df-wdom 7289  df-cnf 7379  df-r1 7452  df-rank 7453  df-card 7588  df-aleph 7589  df-ac 7759  df-cda 7810  df-fin4 7929  df-gch 8259
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