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Theorem alephsucpw2 9522
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 10083 or gchaleph2 10079.) The transposed form alephsucpw 9977 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 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)

Proof of Theorem alephsucpw2
StepHypRef Expression
1 fvex 6664 . . 3 (ℵ‘𝐴) ∈ V
21canth2 8654 . 2 (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
3 alephnbtwn2 9483 . 2 ¬ ((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
42, 3mptnan 1770 1 ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  𝒫 cpw 4520   class class class wbr 5047  suc csuc 6174  cfv 6336  csdm 8491  cale 9349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-inf2 9088
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-se 5496  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-isom 6345  df-riota 7096  df-om 7564  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-oi 8958  df-har 9005  df-card 9352  df-aleph 9353
This theorem is referenced by:  alephsucpw  9977  gchaleph  10078
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