Theorem ackm 3597

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. The equivalence is shown by aceqkm 3596. Maes found this version of AC in April, 2004 (replacing a longer version, also with 5 quantifiers, that he found in November, 2003).

References: http://www.cs.nyu.edu/pipermail/fom/2003-November/007653.html http://www.cs.nyu.edu/pipermail/fom/2003-November/007690.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(s):   x,y,z,v,u

Proof of Theorem ackm

Step	Hyp	Ref	Expression
1		aceqkm 3596	. 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 3569	. 2 ⊢ ∃f(f ⊆ x ∧ f Fn dom x)
3	1, 2	mpgbi 685	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 1   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803   ⊆ wss 1487  dom cdm 2410   Fn wfn 2417

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-ac 1080

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-reu 1207  df-rab 1208  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-iun 1996  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-fr 2169  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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