Theorem ac4 4722

Description: Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4724, derived from this one, shows that this form of the axiom does imply that at least one such set f whose existence we assert is in fact a function. Axiom of Choice of [TakeutiZaring] p. 83.

Takeuti and Zaring call this "weak choice" in contrast to "strong choice" E.FA.z(z =/= (/) -> (F` z) e. z), which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable F and thus cannot be expressed in our first-order formalization. However, it has been shown that ZF plus strong choice is a conservative extension of ZF plus weak choice. See Ulrich Felgner, "Comparison of the axioms of local and universal choice," Fundamenta Mathematica, 71, 43-62 (1971).

Weak choice can be strengthened in a different direction to choose from a collection of proper classes; see ac6s5 4734.

Assertion

Ref	Expression
ac4	|- E.fA.z e. x (z =/= (/) -> (f` z) e. z)

Distinct variable group:   x,z,f

Proof of Theorem ac4

Step	Hyp	Ref	Expression
1		aceq3 4705	. . 3 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z))
2		ac7 4720	. . 3 |- E.f(f (_ x /\ f Fn dom x)
3	1, 2	mpgbi 984	. 2 |- A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z)
4	3	a4i 979	1 |- E.fA.z e. x (z =/= (/) -> (f` z) e. z)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 951   e. wcel 955  E.wex 977   =/= wne 1577  A.wral 1637   (_ wss 2037  (/)c0 2270  dom cdm 3160   Fn wfn 3167  ` cfv 3172

This theorem is referenced by:  ac4c 4723  ac5 4724

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188

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