Theorem ac4 4896

Description: Equivalent of Axiom of Choice. We do not insist that f be a function. However, theorem ac5 4898, 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 4908.

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 4879	. . 3 |- (A.xE.f(f (_ x /\ f Fn dom x) <-> A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z))
2		ac7 4894	. . 3 |- E.f(f (_ x /\ f Fn dom x)
3	1, 2	mpgbi 1023	. 2 |- A.xE.fA.z e. x (z =/= (/) -> (f` z) e. z)
4	3	a4i 1018	1 |- E.fA.z e. x (z =/= (/) -> (f` z) e. z)

Syntax hints:   -> wi 3   /\ wa 221  A.wal 990   e. wcel 994  E.wex 1016   =/= wne 1628  A.wral 1691   (_ wss 2099  (/)c0 2332  dom cdm 3251   Fn wfn 3258  ` cfv 3263

This theorem is referenced by:  ac4c 4897  ac5 4898

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-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-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-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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