Theorem alephom 9407

Description: From canth2 8113, 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 9391 (in the form of cfpwsdom 9406), ( 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

Step	Hyp	Ref	Expression
1		sdomirr 8097	. 2 |- -. om ~< om
2		2onn 7720	. . . . . 6 |- 2o e. om
3	2	elexi 3213	. . . . 5 |- 2o e. _V
4		domrefg 7990	. . . . 5 |- ( 2o e. _V -> 2o ~<_ 2o )
5	3	cfpwsdom 9406	. . . . 5 |- ( 2o ~<_ 2o -> ( aleph ` (/) ) ~< ( cf ` ( card ` ( 2o ^m ( aleph ` (/) ) ) ) ) )
6	3, 4, 5	mp2b 10	. . . 4 |- ( aleph ` (/) ) ~< ( cf ` ( card ` ( 2o ^m ( aleph ` (/) ) ) ) )
7		aleph0 8889	. . . . . 6 |- ( aleph ` (/) ) = om
8	7	a1i 11	. . . . 5 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> ( aleph ` (/) ) = om )
9	7	oveq2i 6661	. . . . . . . . . 10 |- ( 2o ^m ( aleph ` (/) ) ) = ( 2o ^m om )
10	9	fveq2i 6194	. . . . . . . . 9 |- ( card ` ( 2o ^m ( aleph ` (/) ) ) ) = ( card ` ( 2o ^m om ) )
11	10	eqeq1i 2627	. . . . . . . 8 |- ( ( card ` ( 2o ^m ( aleph ` (/) ) ) ) = ( aleph ` om ) <-> ( card ` ( 2o ^m om ) ) = ( aleph ` om ) )
12	11	biimpri 218	. . . . . . 7 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> ( card ` ( 2o ^m ( aleph ` (/) ) ) ) = ( aleph ` om ) )
13	12	fveq2d 6195	. . . . . 6 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> ( cf ` ( card ` ( 2o ^m ( aleph ` (/) ) ) ) ) = ( cf ` ( aleph ` om ) ) )
14		limom 7080	. . . . . . . 8 |- Lim om
15		alephsing 9098	. . . . . . . 8 |- ( Lim om -> ( cf ` ( aleph ` om ) ) = ( cf ` om ) )
16	14, 15	ax-mp 5	. . . . . . 7 |- ( cf ` ( aleph ` om ) ) = ( cf ` om )
17		cfom 9086	. . . . . . 7 |- ( cf ` om ) = om
18	16, 17	eqtri 2644	. . . . . 6 |- ( cf ` ( aleph ` om ) ) = om
19	13, 18	syl6eq 2672	. . . . 5 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> ( cf ` ( card ` ( 2o ^m ( aleph ` (/) ) ) ) ) = om )
20	8, 19	breq12d 4666	. . . 4 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> ( ( aleph ` (/) ) ~< ( cf ` ( card ` ( 2o ^m ( aleph ` (/) ) ) ) ) <-> om ~< om ) )
21	6, 20	mpbii 223	. . 3 |- ( ( card ` ( 2o ^m om ) ) = ( aleph ` om ) -> om ~< om )
22	21	necon3bi 2820	. 2 |- ( -. om ~< om -> ( card ` ( 2o ^m om ) ) =/= ( aleph ` om ) )
23	1, 22	ax-mp 5	1 |- ( card ` ( 2o ^m om ) ) =/= ( aleph ` om )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   Lim wlim 5724   ` cfv 5888  (class class class)co 6650   omcom 7065   2oc2o 7554   ^m cmap 7857   ~<_ cdom 7953   ~< csdm 7954   cardccrd 8761   alephcale 8762   cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-ac2 9285

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768  df-ac 8939

This theorem is referenced by: (None)