Theorem ackm 4799

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 aceqkm 4798. 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.

Assertion

Ref	Expression
ackm	|- A.xE.yA.zE.vA.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

Step	Hyp	Ref	Expression
1		aceqkm 4798	. 2 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.yA.zE.vA.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))))))
2		ac7 4765	. 2 |- E.f(f (_ x /\ f Fn dom x)
3	1, 2	mpgbi 991	1 |- A.xE.yA.zE.vA.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)))))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223  A.wal 958   = wceq 960   e. wcel 962  E.wex 984   (_ wss 2058  dom cdm 3186   Fn wfn 3193

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-rep 2708  ax-sep 2718  ax-nul 2725  ax-pow 2758  ax-pr 2795  ax-un 2882  ax-ac 4761

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-reu 1658  df-rab 1659  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-uni 2518  df-iun 2582  df-br 2635  df-opab 2682  df-id 2851  df-xp 3200  df-rel 3201  df-cnv 3202  df-co 3203  df-dm 3204  df-rn 3205  df-res 3206  df-ima 3207  df-fun 3208  df-fn 3209  df-fv 3214

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