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Theorem gch3 9071
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 461 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  x  e.  On )
2 fvex 5882 . . . . 5  |-  ( aleph `  x )  e.  _V
3 simpl 457 . . . . 5  |-  ( (GCH  =  _V  /\  x  e.  On )  -> GCH  =  _V )
42, 3syl5eleqr 2552 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
5 fvex 5882 . . . . 5  |-  ( aleph ` 
suc  x )  e. 
_V
65, 3syl5eleqr 2552 . . . 4  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
7 gchaleph2 9067 . . . 4  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
81, 4, 6, 7syl3anc 1228 . . 3  |-  ( (GCH  =  _V  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
98ralrimiva 2871 . 2  |-  (GCH  =  _V  ->  A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
10 alephgch 9069 . . . . . 6  |-  ( (
aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ( aleph `  x )  e. GCH )
1110ralimi 2850 . . . . 5  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  A. x  e.  On  ( aleph `  x
)  e. GCH )
12 alephfnon 8463 . . . . . 6  |-  aleph  Fn  On
13 ffnfv 6058 . . . . . 6  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  A. x  e.  On  ( aleph `  x )  e. GCH ) )
1412, 13mpbiran 918 . . . . 5  |-  ( aleph : On -->GCH 
<-> 
A. x  e.  On  ( aleph `  x )  e. GCH )
1511, 14sylibr 212 . . . 4  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  aleph : On -->GCH )
16 df-f 5598 . . . . 5  |-  ( aleph : On -->GCH 
<->  ( aleph  Fn  On  /\  ran  aleph  C_ GCH ) )
1712, 16mpbiran 918 . . . 4  |-  ( aleph : On -->GCH 
<->  ran  aleph  C_ GCH )
1815, 17sylib 196 . . 3  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  ->  ran  aleph  C_ GCH )
19 gch2 9070 . . 3  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
2018, 19sylibr 212 . 2  |-  ( A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
)  -> GCH  =  _V )
219, 20impbii 188 1  |-  (GCH  =  _V 
<-> 
A. x  e.  On  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109    C_ wss 3471   ~Pcpw 4015   class class class wbr 4456   Oncon0 4887   suc csuc 4889   ran crn 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594    ~~ cen 7532   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-reg 8036  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-fal 1401  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-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-seqom 7131  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154  df-er 7329  df-map 7440  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-oi 7953  df-har 8002  df-wdom 8003  df-cnf 8096  df-r1 8199  df-rank 8200  df-card 8337  df-aleph 8338  df-ac 8514  df-cda 8565  df-fin4 8684  df-gch 9016
This theorem is referenced by: (None)
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