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Theorem alephom 8416
Description: From canth2 7219, we know that  (
aleph `  0 )  < 
( 2 ^ om ), but we cannot prove that  ( 2 ^ om )  =  ( aleph `  1 ) (this is the Continuum Hypothesis), nor can we prove that it is less than any bound whatsoever (i.e. the statement  ( aleph `  A )  <  ( 2 ^ om ) is consistent for any ordinal  A). However, we can prove that  ( 2 ^ om ) is not equal to  ( aleph `  om ), nor  ( aleph `  ( aleph `  om ) ), on cofinality grounds, because by Konig's Theorem konigth 8400 (in the form of cfpwsdom 8415), 
( 2 ^ om ) has uncountable cofinality, which eliminates limit alephs like 
( aleph `  om ). (The first limit aleph that is not eliminated is  (
aleph `  ( aleph `  1
) ), which has cofinality  ( aleph `  1 ).) (Contributed by Mario Carneiro, 21-Mar-2013.)
Assertion
Ref Expression
alephom  |-  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om )

Proof of Theorem alephom
StepHypRef Expression
1 sdomirr 7203 . 2  |-  -.  om  ~<  om
2 2onn 6842 . . . . . 6  |-  2o  e.  om
32elexi 2925 . . . . 5  |-  2o  e.  _V
4 domrefg 7101 . . . . 5  |-  ( 2o  e.  _V  ->  2o  ~<_  2o )
53cfpwsdom 8415 . . . . 5  |-  ( 2o  ~<_  2o  ->  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) ) )
63, 4, 5mp2b 10 . . . 4  |-  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )
7 aleph0 7903 . . . . . 6  |-  ( aleph `  (/) )  =  om
87a1i 11 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( aleph `  (/) )  =  om )
97oveq2i 6051 . . . . . . . . . 10  |-  ( 2o 
^m  ( aleph `  (/) ) )  =  ( 2o  ^m  om )
109fveq2i 5690 . . . . . . . . 9  |-  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( card `  ( 2o  ^m  om ) )
1110eqeq1i 2411 . . . . . . . 8  |-  ( (
card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om )  <->  ( card `  ( 2o  ^m  om ) )  =  (
aleph `  om ) )
1211biimpri 198 . . . . . . 7  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( aleph ` 
om ) )
1312fveq2d 5691 . . . . . 6  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  ( cf `  ( aleph ` 
om ) ) )
14 limom 4819 . . . . . . . 8  |-  Lim  om
15 alephsing 8112 . . . . . . . 8  |-  ( Lim 
om  ->  ( cf `  ( aleph `  om ) )  =  ( cf `  om ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( cf `  ( aleph `  om ) )  =  ( cf `  om )
17 cfom 8100 . . . . . . 7  |-  ( cf ` 
om )  =  om
1816, 17eqtri 2424 . . . . . 6  |-  ( cf `  ( aleph `  om ) )  =  om
1913, 18syl6eq 2452 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  om )
208, 19breq12d 4185 . . . 4  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( ( aleph `  (/) )  ~< 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  <->  om  ~<  om )
)
216, 20mpbii 203 . . 3  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  ->  om  ~<  om )
2221necon3bi 2608 . 2  |-  ( -. 
om  ~<  om  ->  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om ) )
231, 22ax-mp 8 1  |-  ( card `  ( 2o  ^m  om ) )  =/=  ( aleph `  om )
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916   (/)c0 3588   class class class wbr 4172   Lim wlim 4542   omcom 4804   ` cfv 5413  (class class class)co 6040   2oc2o 6677    ^m cmap 6977    ~<_ cdom 7066    ~< csdm 7067   cardccrd 7778   alephcale 7779   cfccf 7780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-ac2 8299
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-smo 6567  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-har 7482  df-card 7782  df-aleph 7783  df-cf 7784  df-acn 7785  df-ac 7953
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