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Theorem ac4 8118
Description: Equivalent of Axiom of Choice. We do not insist that  f be a function. However, theorem ac5 8120, 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. F A. 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 8134. (Contributed by NM, 21-Jul-1996.)

Assertion
Ref Expression
ac4  |-  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z )
Distinct variable group:    x, z, f

Proof of Theorem ac4
StepHypRef Expression
1 dfac3 7764 . 2  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
21axaci 8111 1  |-  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    e. wcel 1696    =/= wne 2459   A.wral 2556   (/)c0 3468   ` cfv 5271
This theorem is referenced by:  ac4c  8119
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-ac2 8105
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ac 7759
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