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Theorem gch-kn 8603
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 8507 to the successor aleph using enen2 7297. (Contributed by NM, 1-Oct-2004.)
Assertion
Ref Expression
gch-kn
Distinct variable group:   ,

Proof of Theorem gch-kn
StepHypRef Expression
1 alephexp2 8507 . . 3
2 enen2 7297 . . 3
31, 2syl 16 . 2
43bicomd 194 1
Colors of variables: wff set class
Syntax hints:  ->wi 4  <->wb 178  /\wa 360  e.wcel 1728  {cab 2429  C_wss 3309   class class class wbr 4243   con0 4622  succsuc 4624  `cfv 5501  (class class class)co 6129   c2o 6767   cmap 7067   cen 7155   cale 7874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-13 1730  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-rep 4354  ax-sep 4364  ax-nul 4372  ax-pow 4416  ax-pr 4442  ax-un 4742  ax-inf2 7645  ax-ac2 8394
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-reu 2719  df-rmo 2720  df-rab 2721  df-v 2967  df-sbc 3171  df-csb 3271  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-pss 3325  df-nul 3617  df-if 3766  df-pw 3828  df-sn 3847  df-pr 3848  df-tp 3849  df-op 3850  df-uni 4044  df-int 4080  df-iun 4124  df-br 4244  df-opab 4302  df-mpt 4303  df-tr 4337  df-eprel 4535  df-id 4539  df-po 4544  df-so 4545  df-fr 4582  df-se 4583  df-we 4584  df-ord 4625  df-on 4626  df-lim 4627  df-suc 4628  df-om 4887  df-xp 4925  df-rel 4926  df-cnv 4927  df-co 4928  df-dm 4929  df-rn 4930  df-res 4931  df-ima 4932  df-iota 5464  df-fun 5503  df-fn 5504  df-f 5505  df-f1 5506  df-fo 5507  df-f1o 5508  df-fv 5509  df-isom 5510  df-ov 6132  df-oprab 6133  df-mpt2 6134  df-1st 6399  df-2nd 6400  df-riota 6599  df-recs 6682  df-rdg 6717  df-1o 6773  df-2o 6774  df-oadd 6777  df-er 6954  df-map 7069  df-en 7159  df-dom 7160  df-sdom 7161  df-fin 7162  df-oi 7528  df-har 7575  df-card 7877  df-aleph 7878  df-acn 7880  df-ac 8048
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