Theorem ac4 4733

Description: Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4735, 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 4745.

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 4716	. . 3 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z))
2		ac7 4731	. . 3 |- E.f(f (_ x /\ f Fn dom x)
3	1, 2	mpgbi 986	. 2 |- A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z)
4	3	a4i 981	1 |- E.fA.z e. x (z =/= (/) -> (f` z) e. z)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 953   e. wcel 957  E.wex 979   =/= wne 1583  A.wral 1643   (_ wss 2044  (/)c0 2277  dom cdm 3166   Fn wfn 3173  ` cfv 3178

This theorem is referenced by:  ac4c 4734  ac5 4735

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-rep 2689  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862  ax-ac 4727

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-reu 1649  df-rab 1650  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-fv 3194

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