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

Theorem ac4 7986
 Description: Equivalent of Axiom of Choice. We do not insist that be a function. However, theorem ac5 7988, derived from this one, shows that this form of the axiom does imply that at least one such set 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" , which asserts the existence of a universal choice function but requires second-order quantification on (proper) class variable 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 8002. (Contributed by NM, 21-Jul-1996.)
Assertion
Ref Expression
ac4
Distinct variable group:   ,,

Proof of Theorem ac4
StepHypRef Expression
1 dfac3 7632 . 2 CHOICE
21axaci 7979 1
 Colors of variables: wff set class Syntax hints:   wi 6  wex 1537   wcel 1621   wne 2412  wral 2509  c0 3362  cfv 4592 This theorem is referenced by:  ac4c  7987 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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-ac2 7973 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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ac 7627
 Copyright terms: Public domain W3C validator