MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  gch2 Structured version   Visualization version   Unicode version

Theorem gch2 9100
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 3452 . . 3  |-  ran  aleph  C_  _V
2 sseq2 3454 . . 3  |-  (GCH  =  _V  ->  ( ran  aleph  C_ GCH  <->  ran  aleph  C_  _V ) )
31, 2mpbiri 237 . 2  |-  (GCH  =  _V  ->  ran  aleph  C_ GCH )
4 cardidm 8393 . . . . . . . 8  |-  ( card `  ( card `  x
) )  =  (
card `  x )
5 iscard3 8524 . . . . . . . 8  |-  ( (
card `  ( card `  x ) )  =  ( card `  x
)  <->  ( card `  x
)  e.  ( om  u.  ran  aleph ) )
64, 5mpbi 212 . . . . . . 7  |-  ( card `  x )  e.  ( om  u.  ran  aleph )
7 elun 3574 . . . . . . 7  |-  ( (
card `  x )  e.  ( om  u.  ran  aleph
)  <->  ( ( card `  x )  e.  om  \/  ( card `  x
)  e.  ran  aleph ) )
86, 7mpbi 212 . . . . . 6  |-  ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)
9 fingch 9048 . . . . . . . . 9  |-  Fin  C_ GCH
10 nnfi 7765 . . . . . . . . 9  |-  ( (
card `  x )  e.  om  ->  ( card `  x )  e.  Fin )
119, 10sseldi 3430 . . . . . . . 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 3426 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e.  ran  aleph  ->  ( card `  x
)  e. GCH ) )
1412, 13jaod 382 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  ( ( (
card `  x )  e.  om  \/  ( card `  x )  e.  ran  aleph
)  ->  ( card `  x )  e. GCH )
)
158, 14mpi 20 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( card `  x
)  e. GCH )
16 vex 3048 . . . . . . 7  |-  x  e. 
_V
17 alephon 8500 . . . . . . . . . . 11  |-  ( aleph ` 
suc  x )  e.  On
18 simpr 463 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  x  e.  On )
19 simpl 459 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ran  aleph  C_ GCH )
20 alephfnon 8496 . . . . . . . . . . . . . 14  |-  aleph  Fn  On
21 fnfvelrn 6019 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2220, 18, 21sylancr 669 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. 
ran  aleph )
2319, 22sseldd 3433 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  x )  e. GCH )
24 suceloni 6640 . . . . . . . . . . . . . . 15  |-  ( x  e.  On  ->  suc  x  e.  On )
2524adantl 468 . . . . . . . . . . . . . 14  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  suc  x  e.  On )
26 fnfvelrn 6019 . . . . . . . . . . . . . 14  |-  ( (
aleph  Fn  On  /\  suc  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2720, 25, 26sylancr 669 . . . . . . . . . . . . 13  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e.  ran  aleph )
2819, 27sseldd 3433 . . . . . . . . . . . 12  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x )  e. GCH )
29 gchaleph2 9097 . . . . . . . . . . . 12  |-  ( ( x  e.  On  /\  ( aleph `  x )  e. GCH  /\  ( aleph `  suc  x )  e. GCH )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
3018, 23, 28, 29syl3anc 1268 . . . . . . . . . . 11  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )
31 isnumi 8380 . . . . . . . . . . 11  |-  ( ( ( aleph `  suc  x )  e.  On  /\  ( aleph `  suc  x ) 
~~  ~P ( aleph `  x
) )  ->  ~P ( aleph `  x )  e.  dom  card )
3217, 30, 31sylancr 669 . . . . . . . . . 10  |-  ( ( ran  aleph  C_ GCH  /\  x  e.  On )  ->  ~P ( aleph `  x )  e.  dom  card )
3332ralrimiva 2802 . . . . . . . . 9  |-  ( ran  aleph 
C_ GCH  ->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
34 dfac12 8579 . . . . . . . . 9  |-  (CHOICE  <->  A. x  e.  On  ~P ( aleph `  x )  e.  dom  card )
3533, 34sylibr 216 . . . . . . . 8  |-  ( ran  aleph 
C_ GCH  -> CHOICE
)
36 dfac10 8567 . . . . . . . 8  |-  (CHOICE  <->  dom  card  =  _V )
3735, 36sylib 200 . . . . . . 7  |-  ( ran  aleph 
C_ GCH  ->  dom  card  =  _V )
3816, 37syl5eleqr 2536 . . . . . 6  |-  ( ran  aleph 
C_ GCH  ->  x  e.  dom  card )
39 cardid2 8387 . . . . . 6  |-  ( x  e.  dom  card  ->  (
card `  x )  ~~  x )
40 engch 9053 . . . . . 6  |-  ( (
card `  x )  ~~  x  ->  ( (
card `  x )  e. GCH  <-> 
x  e. GCH ) )
4138, 39, 403syl 18 . . . . 5  |-  ( ran  aleph 
C_ GCH  ->  ( ( card `  x )  e. GCH  <->  x  e. GCH ) )
4215, 41mpbid 214 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e. GCH )
4316a1i 11 . . . 4  |-  ( ran  aleph 
C_ GCH  ->  x  e.  _V )
4442, 432thd 244 . . 3  |-  ( ran  aleph 
C_ GCH  ->  ( x  e. GCH  <->  x  e.  _V ) )
4544eqrdv 2449 . 2  |-  ( ran  aleph 
C_ GCH  -> GCH  =  _V )
463, 45impbii 191 1  |-  (GCH  =  _V 
<->  ran  aleph  C_ GCH )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887   A.wral 2737   _Vcvv 3045    u. cun 3402    C_ wss 3404   ~Pcpw 3951   class class class wbr 4402   dom cdm 4834   ran crn 4835   Oncon0 5423   suc csuc 5425    Fn wfn 5577   ` cfv 5582   omcom 6692    ~~ cen 7566   Fincfn 7569   cardccrd 8369   alephcale 8370  CHOICEwac 8546  GCHcgch 9045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-reg 8107  ax-inf2 8146
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-supp 6915  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-seqom 7165  df-1o 7182  df-2o 7183  df-oadd 7186  df-omul 7187  df-oexp 7188  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-fsupp 7884  df-oi 8025  df-har 8073  df-wdom 8074  df-cnf 8167  df-r1 8235  df-rank 8236  df-card 8373  df-aleph 8374  df-ac 8547  df-cda 8598  df-fin4 8717  df-gch 9046
This theorem is referenced by:  gch3  9101
  Copyright terms: Public domain W3C validator