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Theorem aleph1 8448
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 6726 . . 3  |-  1o  =  suc  (/)
21fveq2i 5733 . 2  |-  ( aleph `  1o )  =  (
aleph `  suc  (/) )
3 alephsucpw 8447 . . 3  |-  ( aleph ` 
suc  (/) )  ~<_  ~P ( aleph `  (/) )
4 fvex 5744 . . . . 5  |-  ( aleph `  (/) )  e.  _V
54pw2en 7217 . . . 4  |-  ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )
6 domen2 7252 . . . 4  |-  ( ~P ( aleph `  (/) )  ~~  ( 2o  ^m  ( aleph `  (/) ) )  -> 
( ( aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) ) )
75, 6ax-mp 8 . . 3  |-  ( (
aleph `  suc  (/) )  ~<_  ~P ( aleph `  (/) )  <->  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) ) )
83, 7mpbi 201 . 2  |-  ( aleph ` 
suc  (/) )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
92, 8eqbrtri 4233 1  |-  ( aleph `  1o )  ~<_  ( 2o 
^m  ( aleph `  (/) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   (/)c0 3630   ~Pcpw 3801   class class class wbr 4214   suc csuc 4585   ` cfv 5456  (class class class)co 6083   1oc1o 6719   2oc2o 6720    ^m cmap 7020    ~~ cen 7108    ~<_ cdom 7109   alephcale 7825
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-ac2 8345
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-se 4544  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-isom 5465  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-recs 6635  df-rdg 6670  df-1o 6726  df-2o 6727  df-er 6907  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-fin 7115  df-oi 7481  df-har 7528  df-card 7828  df-aleph 7829  df-ac 7999
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