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Theorem alephsucpw2 7993
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 8556 or gchaleph2 8552.) The transposed form alephsucpw 8446 cannot be proven without the AC, and is in fact equlvalent 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 5743 . . 3  |-  ( aleph `  A )  e.  _V
21canth2 7261 . 2  |-  ( aleph `  A )  ~<  ~P ( aleph `  A )
3 alephnbtwn2 7954 . 2  |-  -.  (
( aleph `  A )  ~<  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) )
42, 3mpto1 1543 1  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3   ~Pcpw 3800   class class class wbr 4213   suc csuc 4584   ` cfv 5455    ~< csdm 7109   alephcale 7824
This theorem is referenced by:  alephsucpw  8446  gchaleph  8551
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-oi 7480  df-har 7527  df-card 7827  df-aleph 7828
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