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Theorem gch2 8956
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 3487 . . 3  |-  ran  aleph  C_  _V
2 sseq2 3489 . . 3  |-  (GCH  =  _V  ->  ( ran  aleph  C_ GCH  <->  ran  aleph  C_  _V ) )
31, 2mpbiri 233 . 2  |-  (GCH  =  _V  ->  ran  aleph  C_ GCH )
4 cardidm 8243 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
5 iscard3 8377 . . . . . . . 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 3608 . . . . . . 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 8904 . . . . . . . . 9  |-  Fin  C_ GCH
10 nnfi 7617 . . . . . . . . 9  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
119, 10sseldi 3465 . . . . . . . 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 3461 . . . . . . 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 3081 . . . . . . 7  |-  x  e. 
_V
17 alephon 8353 . . . . . . . . . . 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 8349 . . . . . . . . . . . . . 14  |-  aleph  Fn  On
21 fnfvelrn 5952 . . . . . . . . . . . . . 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 3468 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
24 suceloni 6537 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  suc  x  e.  On )
2524adantl 466 . . . . . . . . . . . . . 14  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  suc  x  e.  On )
26 fnfvelrn 5952 . . . . . . . . . . . . . 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 3468 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
29 gchaleph2 8953 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
3018, 23, 28, 29syl3anc 1219 . . . . . . . . . . 11  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
31 isnumi 8230 . . . . . . . . . . 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 2830 . . . . . . . . 9  |-  ( ran  aleph 
C_ GCH  ->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
34 dfac12 8432 . . . . . . . . 9  |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
3533, 34sylibr 212 . . . . . . . 8  |-  ( ran  aleph 
C_ GCH  -> CHOICE
)
36 dfac10 8420 . . . . . . . 8  |-  (CHOICE  <->  dom  card  =  _V )
3735, 36sylib 196 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  dom  card  =  _V )
3816, 37syl5eleqr 2549 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  x  e.  dom  card )
39 cardid2 8237 . . . . . 6  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
40 engch 8909 . . . . . 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 2451 . 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 1370    e. wcel 1758   A.wral 2799   _Vcvv 3078    u. cun 3437    C_ wss 3439   ~Pcpw 3971   class class class wbr 4403   Oncon0 4830   suc csuc 4832   dom cdm 4951   ran crn 4952    Fn wfn 5524   ` cfv 5529   omcom 6589    ~~ cen 7420   Fincfn 7423   cardccrd 8219   alephcale 8220  CHOICEwac 8399  GCHcgch 8901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-reg 7921  ax-inf2 7961
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-supp 6804  df-recs 6945  df-rdg 6979  df-seqom 7016  df-1o 7033  df-2o 7034  df-oadd 7037  df-omul 7038  df-oexp 7039  df-er 7214  df-map 7329  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-fsupp 7735  df-oi 7838  df-har 7887  df-wdom 7888  df-cnf 7982  df-r1 8085  df-rank 8086  df-card 8223  df-aleph 8224  df-ac 8400  df-cda 8451  df-fin4 8570  df-gch 8902
This theorem is referenced by:  gch3  8957
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