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Theorem ackm 7976
Description: A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to 
e.,  = primitives in prenex form), discovered and proved by Kurt Maes. This establishes a new record, reducing from 6 to 5 the largest number of quantified variables needed by any ZFC axiom. The ZF-equivalence to AC is shown by theorem dfackm 7676. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003). See Kurt Maes, "A 5-quantifier ( e.,=)-expression ZF-equivalent to the Axiom of Choice" (http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.3162v1.pdf).

The original FOM posts are: http://www.cs.nyu.edu/pipermail/fom/2003-November/007631.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007641.html. (Contributed by NM, 29-Apr-2004.) (Revised by Mario Carneiro, 17-May-2015.) (Proof modification is discouraged.)

Assertion
Ref Expression
ackm  |-  A. x E. y A. z E. v A. u ( ( y  e.  x  /\  ( z  e.  y  ->  ( ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v
) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) ) )
Distinct variable group:    x, y, z, v, u

Proof of Theorem ackm
StepHypRef Expression
1 axac3 7974 . 2  |- CHOICE
2 dfackm 7676 . 2  |-  (CHOICE  <->  A. x E. y A. z E. v A. u ( ( y  e.  x  /\  ( z  e.  y  ->  ( ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v
) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) ) ) )
31, 2mpbi 201 1  |-  A. x E. y A. z E. v A. u ( ( y  e.  x  /\  ( z  e.  y  ->  ( ( v  e.  x  /\  -.  y  =  v )  /\  z  e.  v
) ) )  \/  ( -.  y  e.  x  /\  ( z  e.  x  ->  (
( v  e.  z  /\  v  e.  y )  /\  ( ( u  e.  z  /\  u  e.  y )  ->  u  =  v ) ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360   A.wal 1532   E.wex 1537    = wceq 1619    e. wcel 1621  CHOICEwac 7626
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ac 7627
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