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Theorem gch-kn 9258
Description: The equivalence of two versions of the Generalized Continuum Hypothesis. The right-hand side is the standard version in the literature. The left-hand side is a version devised by Kannan Nambiar, which he calls the Axiom of Combinatorial Sets. For the notation and motivation behind this axiom, see his paper, "Derivation of Continuum Hypothesis from Axiom of Combinatorial Sets," available at http://www.e-atheneum.net/science/derivation_ch.pdf. The equivalence of the two sides provides a negative answer to Open Problem 2 in http://www.e-atheneum.net/science/open_problem_print.pdf. The key idea in the proof below is to equate both sides of alephexp2 9162 to the successor aleph using enen2 7866. (Contributed by NM, 1-Oct-2004.)
Assertion
Ref Expression
gch-kn (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))))
Distinct variable group:   𝑥,𝐴

Proof of Theorem gch-kn
StepHypRef Expression
1 alephexp2 9162 . . 3 (𝐴 ∈ On → (2𝑜𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
2 enen2 7866 . . 3 ((2𝑜𝑚 (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} → ((ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
31, 2syl 17 . 2 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
43bicomd 211 1 (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2𝑜𝑚 (ℵ‘𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1938  {cab 2500  wss 3444   class class class wbr 4481  Oncon0 5530  suc csuc 5532  cfv 5694  (class class class)co 6431  2𝑜c2o 7321  𝑚 cmap 7624  cen 7718  cale 8525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6728  ax-inf2 8301  ax-ac2 9048
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5658  df-fun 5696  df-fn 5697  df-f 5698  df-f1 5699  df-fo 5700  df-f1o 5701  df-fv 5702  df-isom 5703  df-riota 6393  df-ov 6434  df-oprab 6435  df-mpt2 6436  df-om 6839  df-1st 6939  df-2nd 6940  df-wrecs 7174  df-recs 7235  df-rdg 7273  df-1o 7327  df-2o 7328  df-oadd 7331  df-er 7509  df-map 7626  df-en 7722  df-dom 7723  df-sdom 7724  df-fin 7725  df-oi 8178  df-har 8226  df-card 8528  df-aleph 8529  df-acn 8531  df-ac 8702
This theorem is referenced by: (None)
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