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Theorem gch-kn 8545
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 8445 to the successor aleph using enen2 7239. (Contributed by NM, 1-Oct-2004.)
Assertion
Ref Expression
gch-kn  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A
) ) }  <->  ( aleph ` 
suc  A )  ~~  ( 2o  ^m  ( aleph `  A ) ) ) )
Distinct variable group:    x, A

Proof of Theorem gch-kn
StepHypRef Expression
1 alephexp2 8445 . . 3  |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A
) )  ~~  {
x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
2 enen2 7239 . . 3  |-  ( ( 2o  ^m  ( aleph `  A ) )  ~~  { x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) }  ->  (
( aleph `  suc  A ) 
~~  ( 2o  ^m  ( aleph `  A )
)  <->  ( aleph `  suc  A )  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } ) )
31, 2syl 16 . 2  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
~~  ( 2o  ^m  ( aleph `  A )
)  <->  ( aleph `  suc  A )  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } ) )
43bicomd 193 1  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A
) ) }  <->  ( aleph ` 
suc  A )  ~~  ( 2o  ^m  ( aleph `  A ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1725   {cab 2421    C_ wss 3312   class class class wbr 4204   Oncon0 4573   suc csuc 4575   ` cfv 5445  (class class class)co 6072   2oc2o 6709    ^m cmap 7009    ~~ cen 7097   alephcale 7812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-inf2 7585  ax-ac2 8332
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-oi 7468  df-har 7515  df-card 7815  df-aleph 7816  df-acn 7818  df-ac 7986
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