Theorem ackm 4928

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 4927. 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 4927	. 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 4894	. 2 ⊢ ∃f(f ⊆ x ⋀ f Fn dom x)
3	1, 2	mpgbi 1023	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 220   ⋀ wa 221  ∀wal 990   = wceq 992   ∈ wcel 994  ∃wex 1016   ⊆ wss 2099  dom cdm 3251   Fn wfn 3258

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-ac 4890

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-iun 2635  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-fv 3279

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