Theorem alephsucpw2 9994

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 10559 or gchaleph2 10555.) The transposed form alephsucpw 10453 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 6830	. . 3 ⊢ (ℵ‘𝐴) ∈ V
2	1	canth2 9038	. 2 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
3		alephnbtwn2 9955	. 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 4548   class class class wbr 5089  suc csuc 6304  ‘cfv 6477   ≺ csdm 8863  ℵcale 9821

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 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663  ax-inf2 9526

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 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-isom 6486  df-riota 7298  df-ov 7344  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-er 8617  df-en 8865  df-dom 8866  df-sdom 8867  df-fin 8868  df-oi 9391  df-har 9438  df-card 9824  df-aleph 9825

This theorem is referenced by:  alephsucpw  10453  gchaleph  10554