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Theorem alephsucpw2 8391
Description: The power set of an aleph is not strictly dominated by the successor aleph. (The Generalized Continuum Hypothesis says they are equinumerous, see gch3 8953 or gchaleph2 8949.) The transposed form alephsucpw 8844 cannot be proven without the AC, and is in fact equivalent to it. (Contributed by Mario Carneiro, 2-Feb-2013.)
Assertion
Ref Expression
alephsucpw2  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )

Proof of Theorem alephsucpw2
StepHypRef Expression
1 fvex 5808 . . 3  |-  ( aleph `  A )  e.  _V
21canth2 7573 . 2  |-  ( aleph `  A )  ~<  ~P ( aleph `  A )
3 alephnbtwn2 8352 . 2  |-  -.  (
( aleph `  A )  ~<  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) )
42, 3mptnan 1576 1  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   ~Pcpw 3967   class class class wbr 4399   suc csuc 4828   ` cfv 5525    ~< csdm 7418   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
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-om 6586  df-recs 6941  df-rdg 6975  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-oi 7834  df-har 7883  df-card 8219  df-aleph 8220
This theorem is referenced by:  alephsucpw  8844  gchaleph  8948
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