Theorem ackm 4706

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 4705. 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 4705	. 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 4672	. 2 |- E.f(f (_ x /\ f Fn dom x)
3	1, 2	mpgbi 963	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 950  E.wex 956   = wceq 1099   e. wcel 1105   (_ wss 2018  dom cdm 3133   Fn wfn 3140

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830  ax-ac 4668

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-reu 1627  df-rab 1628  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-iun 2536  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161

Copyright terms: Public domain