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Theorem gch2 9065
Description: It is sufficient to require that all alephs are GCH-sets to ensure the full generalized continuum hypothesis. (The proof uses the Axiom of Regularity.) (Contributed by Mario Carneiro, 15-May-2015.)
Assertion
Ref Expression
gch2  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )

Proof of Theorem gch2
StepHypRef Expression
1 ssv 3529 . . 3  |-  ran  aleph  C_  _V
2 sseq2 3531 . . 3  |-  (GCH  =  _V  ->  ( ran  aleph  C_ GCH  <->  ran  aleph  C_  _V ) )
31, 2mpbiri 233 . 2  |-  (GCH  =  _V  ->  ran  aleph  C_ GCH )
4 cardidm 8352 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
5 iscard3 8486 . . . . . . . 8  |-  ( (
card `  ( card `  x ) )  =  ( card `  x
)  <->  ( card `  x
)  e.  ( om  u.  ran  aleph ) )
64, 5mpbi 208 . . . . . . 7  |-  ( card `  x )  e.  ( om  u.  ran  aleph )
7 elun 3650 . . . . . . 7  |-  ( (
card `  x )  e.  ( om  u.  ran  aleph
)  <->  ( ( card `  x )  e.  om  \/  ( card `  x
)  e.  ran  aleph ) )
86, 7mpbi 208 . . . . . 6  |-  ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)
9 fingch 9013 . . . . . . . . 9  |-  Fin  C_ GCH
10 nnfi 7722 . . . . . . . . 9  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
119, 10sseldi 3507 . . . . . . . 8  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e. GCH )
1211a1i 11 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  om  ->  ( card `  x
)  e. GCH ) )
13 ssel 3503 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  ran  aleph  ->  ( card `  x
)  e. GCH ) )
1412, 13jaod 380 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  ( ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)  ->  ( card `  x )  e. GCH )
)
158, 14mpi 17 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( card `  x
)  e. GCH )
16 vex 3121 . . . . . . 7  |-  x  e. 
_V
17 alephon 8462 . . . . . . . . . . 11  |-  ( aleph ` 
suc  x )  e.  On
18 simpr 461 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  x  e.  On )
19 simpl 457 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ran  aleph  C_ GCH )
20 alephfnon 8458 . . . . . . . . . . . . . 14  |-  aleph  Fn  On
21 fnfvelrn 6029 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2220, 18, 21sylancr 663 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2319, 22sseldd 3510 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
24 suceloni 6643 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  suc  x  e.  On )
2524adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  suc  x  e.  On )
26 fnfvelrn 6029 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  suc  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2720, 25, 26sylancr 663 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2819, 27sseldd 3510 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
29 gchaleph2 9062 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
3018, 23, 28, 29syl3anc 1228 . . . . . . . . . . 11  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
31 isnumi 8339 . . . . . . . . . . 11  |-  ( ( ( aleph `  suc  x )  e.  On  /\  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )  ->  ~P ( aleph `  x )  e.  dom  card )
3217, 30, 31sylancr 663 . . . . . . . . . 10  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ~P ( aleph `  x )  e.  dom  card )
3332ralrimiva 2881 . . . . . . . . 9  |-  ( ran  aleph 
C_ GCH  ->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
34 dfac12 8541 . . . . . . . . 9  |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
3533, 34sylibr 212 . . . . . . . 8  |-  ( ran  aleph 
C_ GCH  -> CHOICE
)
36 dfac10 8529 . . . . . . . 8  |-  (CHOICE  <->  dom  card  =  _V )
3735, 36sylib 196 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  dom  card  =  _V )
3816, 37syl5eleqr 2562 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  x  e.  dom  card )
39 cardid2 8346 . . . . . 6  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
40 engch 9018 . . . . . 6  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  e. GCH  <-> 
x  e. GCH ) )
4138, 39, 403syl 20 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e. GCH  <->  x  e. GCH ) )
4215, 41mpbid 210 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e. GCH )
4316a1i 11 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e.  _V )
4442, 432thd 240 . . 3  |-  ( ran  aleph 
C_ GCH  ->  ( x  e. GCH  <->  x  e.  _V ) )
4544eqrdv 2464 . 2  |-  ( ran  aleph 
C_ GCH  -> GCH  =  _V )
463, 45impbii 188 1  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2817   _Vcvv 3118    u. cun 3479    C_ wss 3481   ~Pcpw 4016   class class class wbr 4453   Oncon0 4884   suc csuc 4886   dom cdm 5005   ran crn 5006    Fn wfn 5589   ` cfv 5594   omcom 6695    ~~ cen 7525   Fincfn 7528   cardccrd 8328   alephcale 8329  CHOICEwac 8508  GCHcgch 9010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-reg 8030  ax-inf2 8070
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-seqom 7125  df-1o 7142  df-2o 7143  df-oadd 7146  df-omul 7147  df-oexp 7148  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-oi 7947  df-har 7996  df-wdom 7997  df-cnf 8091  df-r1 8194  df-rank 8195  df-card 8332  df-aleph 8333  df-ac 8509  df-cda 8560  df-fin4 8679  df-gch 9011
This theorem is referenced by:  gch3  9066
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