MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac4 Unicode version

Theorem ac4 8056
Description: Equivalent of Axiom of Choice. We do not insist that  f be a function. However, theorem ac5 8058, 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 8072. (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 7702 . 2  |-  (CHOICE  <->  A. x E. f A. z  e.  x  ( z  =/=  (/)  ->  ( f `  z )  e.  z ) )
21axaci 8049 1  |-  E. f A. z  e.  x  ( z  =/=  (/)  ->  (
f `  z )  e.  z )
Colors of variables: wff set class
Syntax hints:    -> wi 6   E.wex 1537    e. wcel 1621    =/= wne 2419   A.wral 2516   (/)c0 3416   ` cfv 4659
This theorem is referenced by:  ac4c  8057
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-ac2 8043
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ac 7697
  Copyright terms: Public domain W3C validator