Theorem gch-kn 10743

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 10647 to the successor aleph using enen2 9121. (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) ≈ (2o ↑m (ℵ‘𝐴))))

Distinct variable group:   𝑥,𝐴

Proof of Theorem gch-kn

Step	Hyp	Ref	Expression
1		alephexp2 10647	. . 3 ⊢ (𝐴 ∈ On → (2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))})
2		enen2 9121	. . 3 ⊢ ((2o ↑m (ℵ‘𝐴)) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} → ((ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
3	1, 2	syl 18	. 2 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴)) ↔ (ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))}))
4	3	bicomd 226	1 ⊢ (𝐴 ∈ On → ((ℵ‘suc 𝐴) ≈ {𝑥 ∣ (𝑥 ⊆ (ℵ‘𝐴) ∧ 𝑥 ≈ (ℵ‘𝐴))} ↔ (ℵ‘suc 𝐴) ≈ (2o ↑m (ℵ‘𝐴))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  {cab 2739   ⊆ wss 3899   class class class wbr 5103  Oncon0 6355  suc csuc 6357  ‘cfv 6531  (class class class)co 7412  2_oc2o 8454   ↑_m cmap 8831   ≈ cen 8954  ℵcale 9998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740  ax-inf2 9626  ax-ac2 10522

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8460  df-2o 8461  df-er 8701  df-map 8833  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9488  df-har 9535  df-card 10001  df-aleph 10002  df-acn 10004  df-ac 10176

This theorem is used by: (None)