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Theorem aleph1 8845
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 7029 . . 3  |-  1o  =  suc  (/)
21fveq2i 5801 . 2  |-  ( aleph `  1o )  =  (
aleph `  suc  (/) )
3 alephsucpw 8844 . . 3  |-  ( aleph ` 
suc  (/) )  ~<_  ~P ( aleph `  (/) )
4 fvex 5808 . . . . 5  |-  ( aleph `  (/) )  e.  _V
54pw2en 7527 . . . 4  |-  ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )
6 domen2 7563 . . . 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 4418 1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184   (/)c0 3744   ~Pcpw 3967   class class class wbr 4399   suc csuc 4828   ` cfv 5525  (class class class)co 6199   1oc1o 7022   2oc2o 7023    ^m cmap 7323    ~~ cen 7416    ~<_ cdom 7417   alephcale 8216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-inf2 7957  ax-ac2 8742
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-se 4787  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-isom 5534  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-er 7210  df-map 7325  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-oi 7834  df-har 7883  df-card 8219  df-aleph 8220  df-ac 8396
This theorem is referenced by: (None)
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