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Theorem alephom 8386
Description: From canth2 7189, 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 8370 (in the form of cfpwsdom 8385), 
( 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 7173 . 2  |-  -.  om  ~<  om
2 2onn 6812 . . . . . 6  |-  2o  e.  om
32elexi 2901 . . . . 5  |-  2o  e.  _V
4 domrefg 7071 . . . . 5  |-  ( 2o  e.  _V  ->  2o  ~<_  2o )
53cfpwsdom 8385 . . . . 5  |-  ( 2o  ~<_  2o  ->  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) ) )
63, 4, 5mp2b 10 . . . 4  |-  ( aleph `  (/) )  ~<  ( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )
7 aleph0 7873 . . . . . 6  |-  ( aleph `  (/) )  =  om
87a1i 11 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( aleph `  (/) )  =  om )
97oveq2i 6024 . . . . . . . . . 10  |-  ( 2o 
^m  ( aleph `  (/) ) )  =  ( 2o  ^m  om )
109fveq2i 5664 . . . . . . . . 9  |-  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) )  =  ( card `  ( 2o  ^m  om ) )
1110eqeq1i 2387 . . . . . . . 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 5665 . . . . . 6  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  ( cf `  ( aleph ` 
om ) ) )
14 limom 4793 . . . . . . . 8  |-  Lim  om
15 alephsing 8082 . . . . . . . 8  |-  ( Lim 
om  ->  ( cf `  ( aleph `  om ) )  =  ( cf `  om ) )
1614, 15ax-mp 8 . . . . . . 7  |-  ( cf `  ( aleph `  om ) )  =  ( cf `  om )
17 cfom 8070 . . . . . . 7  |-  ( cf ` 
om )  =  om
1816, 17eqtri 2400 . . . . . 6  |-  ( cf `  ( aleph `  om ) )  =  om
1913, 18syl6eq 2428 . . . . 5  |-  ( (
card `  ( 2o  ^m 
om ) )  =  ( aleph `  om )  -> 
( cf `  ( card `  ( 2o  ^m  ( aleph `  (/) ) ) ) )  =  om )
208, 19breq12d 4159 . . . 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 2584 . 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 1717    =/= wne 2543   _Vcvv 2892   (/)c0 3564   class class class wbr 4146   Lim wlim 4516   omcom 4778   ` cfv 5387  (class class class)co 6013   2oc2o 6647    ^m cmap 6947    ~<_ cdom 7036    ~< csdm 7037   cardccrd 7748   alephcale 7749   cfccf 7750
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-ac2 8269
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-smo 6537  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-map 6949  df-ixp 6993  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-oi 7405  df-har 7452  df-card 7752  df-aleph 7753  df-cf 7754  df-acn 7755  df-ac 7923
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