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Theorem aleph1 8727
Description: The set exponentiation of 2 to the aleph-zero has cardinality of at least aleph-one. (If we were to assume the Continuum Hypothesis, their cardinalities would be the same.) (Contributed by NM, 7-Jul-2004.)
Assertion
Ref Expression
aleph1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )

Proof of Theorem aleph1
StepHypRef Expression
1 df-1o 6912 . . 3  |-  1o  =  suc  (/)
21fveq2i 5689 . 2  |-  ( aleph `  1o )  =  (
aleph `  suc  (/) )
3 alephsucpw 8726 . . 3  |-  ( aleph ` 
suc  (/) )  ~<_  ~P ( aleph `  (/) )
4 fvex 5696 . . . . 5  |-  ( aleph `  (/) )  e.  _V
54pw2en 7410 . . . 4  |-  ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )
6 domen2 7446 . . . 4  |-  ( ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )  -> 
( ( aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) ) )
75, 6ax-mp 5 . . 3  |-  ( (
aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) )
83, 7mpbi 208 . 2  |-  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
92, 8eqbrtri 4306 1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   (/)c0 3632   ~Pcpw 3855   class class class wbr 4287   suc csuc 4716   ` cfv 5413  (class class class)co 6086   1oc1o 6905   2oc2o 6906    ^m cmap 7206    ~~ cen 7299    ~<_ cdom 7300   alephcale 8098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-ac2 8624
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  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-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-oi 7716  df-har 7765  df-card 8101  df-aleph 8102  df-ac 8278
This theorem is referenced by: (None)
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