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Theorem alephsucpw2 8483
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 9043 or gchaleph2 9039.) The transposed form alephsucpw 8936 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 5858 . . 3  |-  ( aleph `  A )  e.  _V
21canth2 7663 . 2  |-  ( aleph `  A )  ~<  ~P ( aleph `  A )
3 alephnbtwn2 8444 . 2  |-  -.  (
( aleph `  A )  ~<  ~P ( aleph `  A
)  /\  ~P ( aleph `  A )  ~< 
( aleph `  suc  A ) )
42, 3mptnan 1605 1  |-  -.  ~P ( aleph `  A )  ~<  ( aleph `  suc  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   ~Pcpw 3999   class class class wbr 4439   suc csuc 4869   ` cfv 5570    ~< csdm 7508   alephcale 8308
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-om 6674  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-oi 7927  df-har 7976  df-card 8311  df-aleph 8312
This theorem is referenced by:  alephsucpw  8936  gchaleph  9038
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