Theorem alephsucpw2 10009

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 10574 or gchaleph2 10570.) The transposed form alephsucpw 10468 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

Step	Hyp	Ref	Expression
1		fvex 6841	. . 3 ⊢ (ℵ‘𝐴) ∈ V
2	1	canth2 9050	. 2 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
3		alephnbtwn2 9970	. 2 ⊢ ¬ ((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
4	2, 3	mptnan 1769	1 ⊢ ¬ 𝒫 (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)

Syntax hints:  ¬ wn 3  𝒫 cpw 4549   class class class wbr 5093  suc csuc 6313  ‘cfv 6486   ≺ csdm 8874  ℵcale 9836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-om 7803  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-oi 9403  df-har 9450  df-card 9839  df-aleph 9840

This theorem is referenced by:  alephsucpw  10468  gchaleph  10569