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Theorem alephsucpw 8437
Description: The power set of an aleph dominates the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8547 or gchaleph2 8543.) (Contributed by NM, 27-Aug-2005.)
Assertion
Ref Expression
alephsucpw  |-  ( aleph ` 
suc  A )  ~<_  ~P ( aleph `  A )

Proof of Theorem alephsucpw
StepHypRef Expression
1 alephsucpw2 7984 . 2  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
2 fvex 5734 . . 3  |-  ( aleph ` 
suc  A )  e. 
_V
3 fvex 5734 . . . 4  |-  ( aleph `  A )  e.  _V
43pwex 4374 . . 3  |-  ~P ( aleph `  A )  e. 
_V
5 domtri 8423 . . 3  |-  ( ( ( aleph `  suc  A )  e.  _V  /\  ~P ( aleph `  A )  e.  _V )  ->  (
( aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  <->  -.  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) ) )
62, 4, 5mp2an 654 . 2  |-  ( (
aleph `  suc  A )  ~<_  ~P ( aleph `  A
)  <->  -.  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) )
71, 6mpbir 201 1  |-  ( aleph ` 
suc  A )  ~<_  ~P ( aleph `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    e. wcel 1725   _Vcvv 2948   ~Pcpw 3791   class class class wbr 4204   suc csuc 4575   ` cfv 5446    ~<_ cdom 7099    ~< csdm 7100   alephcale 7815
This theorem is referenced by:  aleph1  8438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-ac2 8335
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-har 7518  df-card 7818  df-aleph 7819  df-ac 7989
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