Theorem gch-kn 4957

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.rci.rutgers.edu/~kannan/science/derivation_ch.pdf . This theorem provides a negative answer to Open Problem 2 in http://www.rci.rutgers.edu/~kannan/science/open_problem_print.pdf . The key idea in the proof below is to equate both sides of alephexp2 4956 to the successor aleph using enen2 3376.

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(s):   x,A

Proof of Theorem gch-kn

Step	Hyp	Ref	Expression
1		alephexp2 4956	. . 3 ⊢ (A ∈ On → (2o ↑m (ℵ ‘A)) ≈ {x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))})
2		df-pw 1799	. . . . . 6 ⊢ ℘(ℵ ‘A) = {x∣x ⊆ (ℵ ‘A)}
3		fvex 2838	. . . . . . 7 ⊢ (ℵ ‘A) ∈ V
4	3	pwex 1806	. . . . . 6 ⊢ ℘(ℵ ‘A) ∈ V
5	2, 4	eqeltrr 1160	. . . . 5 ⊢ {x∣x ⊆ (ℵ ‘A)} ∈ V
6		pm3.26 256	. . . . . 6 ⊢ ((x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A)) → x ⊆ (ℵ ‘A))
7	6	ss2abi 1552	. . . . 5 ⊢ {x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))} ⊆ {x∣x ⊆ (ℵ ‘A)}
8	5, 7	ssexi 1701	. . . 4 ⊢ {x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))} ∈ V
9		enen2 3376	. . . 4 ⊢ (({x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))} ∈ V ∧ (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))}))
10	8, 9	mpan 518	. . 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))}))
11	1, 10	syl 12	. 2 ⊢ (A ∈ On → ((ℵ ‘suc A) ≈ (2o ↑m (ℵ ‘A)) ↔ (ℵ ‘suc A) ≈ {x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))}))
12	11	bicomd 399	1 ⊢ (A ∈ On → ((ℵ ‘suc A) ≈ {x∣(x ⊆ (ℵ ‘A) ∧ x ≈ (ℵ ‘A))} ↔ (ℵ ‘suc A) ≈ (2o ↑m (ℵ ‘A))))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  {cab 1090   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798   class class class wbr 2054  Oncon0 2199  suc csuc 2201   ‘cfv 2422  (class class class)co 3001  2_oc2o 3100   ↑_m cm 3258   ≈ cen 3271  ℵcale 3621

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439  df-rdg 2970  df-opr 3003  df-oprab 3004  df-1st 3087  df-2nd 3088  df-1o 3104  df-2o 3105  df-oadd 3106  df-omul 3107  df-er 3200  df-ec 3202  df-qs 3205  df-map 3259  df-en 3274  df-dom 3275  df-sdom 3276  df-card 3623  df-aleph 3624  df-ni 3794  df-pli 3795  df-mi 3796  df-lti 3797  df-plpq 3829  df-mpq 3830  df-enq 3831  df-nq 3832  df-plq 3833  df-mq 3834  df-rq 3835  df-ltq 3836  df-1q 3837  df-np 3880  df-1p 3881  df-plp 3882  df-mp 3883  df-ltp 3884  df-plpr 3958  df-mpr 3959  df-enr 3960  df-nr 3961  df-plr 3962  df-mr 3963  df-ltr 3964  df-0r 3965  df-1r 3966  df-m1r 3967  df-c 4034  df-0 4035  df-1 4036  df-r 4038  df-plus 4039  df-mul 4040  df-lt 4041  df-sub 4133  df-neg 4135  df-div 4216  df-le 4277  df-n 4423  df-2 4462  df-n0 4535  df-z 4564  df-seq 4661  df-exp 4676

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