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Theorem gch-kn 8303
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 8203 to the successor aleph using enen2 7002. (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 8203 . . 3  |-  ( A  e.  On  ->  ( 2o  ^m  ( aleph `  A
) )  ~~  {
x  |  ( x 
C_  ( aleph `  A
)  /\  x  ~~  ( aleph `  A )
) } )
2 enen2 7002 . . 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 15 . 2  |-  ( A  e.  On  ->  (
( aleph `  suc  A ) 
~~  ( 2o  ^m  ( aleph `  A )
)  <->  ( aleph `  suc  A )  ~~  { x  |  ( x  C_  ( aleph `  A )  /\  x  ~~  ( aleph `  A ) ) } ) )
43bicomd 192 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 176    /\ wa 358    e. wcel 1684   {cab 2269    C_ wss 3152   class class class wbr 4023   Oncon0 4392   suc csuc 4394   ` cfv 5255  (class class class)co 5858   2oc2o 6473    ^m cmap 6772    ~~ cen 6860   alephcale 7569
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-ac2 8089
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-har 7272  df-card 7572  df-aleph 7573  df-acn 7575  df-ac 7743
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