Theorem ackm 4799

Description: A remarkable equivalent to the Axiom of Choice that has only 5 quantifiers (when expanded to ∈, = 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 (∈,=)-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	⊢ ∀x∃y∀z∃v∀u((y ∈ x ⋀ (z ∈ y → ((v ∈ x ⋀ ¬ y = v) ⋀ z ∈ v))) ⋁ (¬ y ∈ x ⋀ (z ∈ x → ((v ∈ z ⋀ v ∈ y) ⋀ ((u ∈ z ⋀ u ∈ y) → u = v)))))

Distinct variable group:   x,y,z,v,u

Proof of Theorem ackm

Step	Hyp	Ref	Expression
1		aceqkm 4798	. 2 ⊢ (∀x∃f(f ⊆ x ⋀ f Fn dom x) ↔ ∀x∃y∀z∃v∀u((y ∈ x ⋀ (z ∈ y → ((v ∈ x ⋀ ¬ y = v) ⋀ z ∈ v))) ⋁ (¬ y ∈ x ⋀ (z ∈ x → ((v ∈ z ⋀ v ∈ y) ⋀ ((u ∈ z ⋀ u ∈ y) → u = v))))))
2		ac7 4765	. 2 ⊢ ∃f(f ⊆ x ⋀ f Fn dom x)
3	1, 2	mpgbi 989	1 ⊢ ∀x∃y∀z∃v∀u((y ∈ x ⋀ (z ∈ y → ((v ∈ x ⋀ ¬ y = v) ⋀ z ∈ v))) ⋁ (¬ y ∈ x ⋀ (z ∈ x → ((v ∈ z ⋀ v ∈ y) ⋀ ((u ∈ z ⋀ u ∈ y) → u = v)))))

Syntax hints:  ¬ wn 2   → wi 3   ⋁ wo 222   ⋀ wa 223  ∀wal 956   = wceq 958   ∈ wcel 960  ∃wex 982   ⊆ wss 2051  dom cdm 3177   Fn wfn 3184

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2699  ax-sep 2709  ax-nul 2716  ax-pow 2749  ax-pr 2786  ax-un 2873  ax-ac 4761

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-dif 2053  df-un 2054  df-in 2055  df-ss 2057  df-nul 2285  df-pw 2407  df-sn 2417  df-pr 2418  df-op 2421  df-uni 2509  df-iun 2573  df-br 2626  df-opab 2673  df-id 2842  df-xp 3191  df-rel 3192  df-cnv 3193  df-co 3194  df-dm 3195  df-rn 3196  df-res 3197  df-ima 3198  df-fun 3199  df-fn 3200  df-fv 3205

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