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Axiom ax-ac 8886
Description: Axiom of Choice. The Axiom of Choice (AC) is usually considered an extension of ZF set theory rather than a proper part of it. It is sometimes considered philosophically controversial because it asserts the existence of a set without telling us what the set is. ZF set theory that includes AC is called ZFC.

The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each nonempty member of  x. One entry in the pair is the member of  x, and the other entry is some arbitrary member of that member of  x. See the rewritten version ac3 8889 for a more detailed explanation. Theorem ac2 8888 shows an equivalent written compactly with restricted quantifiers.

This version was specifically crafted to be short when expanded to primitives. Kurt Maes' 5-quantifier version ackm 8892 is slightly shorter when the biconditional of ax-ac 8886 is expanded into implication and negation. In axac3 8891 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 9103 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8919, ac5 8904, and ac7 8900. The Axiom of Regularity ax-reg 8104 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8558. Equivalents to AC are the well-ordering theorem weth 8922 and Zorn's lemma zorn 8934. See ac4 8902 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 8104 for derivation of AC equivalents, we provide ax-ac2 8890 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8890 from ax-ac 8886 is shown by theorem axac2 8893, and the reverse derivation by axac 8894. Therefore, new proofs should normally use ax-ac2 8890 instead. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)

Assertion
Ref Expression
ax-ac  |-  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x
)  ->  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
) )
Distinct variable group:    x, y, z, w, v, u, t

Detailed syntax breakdown of Axiom ax-ac
StepHypRef Expression
1 vz . . . . . . 7  setvar  z
2 vw . . . . . . 7  setvar  w
31, 2wel 1887 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  setvar  x
52, 4wel 1887 . . . . . 6  wff  w  e.  x
63, 5wa 371 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  setvar  u
87, 2wel 1887 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  setvar  t
102, 9wel 1887 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 371 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1887 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  setvar  y
149, 13wel 1887 . . . . . . . . . . 11  wff  t  e.  y
1512, 14wa 371 . . . . . . . . . 10  wff  ( u  e.  t  /\  t  e.  y )
1611, 15wa 371 . . . . . . . . 9  wff  ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y ) )
1716, 9wex 1662 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  setvar  v
197, 18weq 1790 . . . . . . . 8  wff  u  =  v
2017, 19wb 188 . . . . . . 7  wff  ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
2120, 7wal 1441 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1662 . . . . 5  wff  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
236, 22wi 4 . . . 4  wff  ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v ) )
2423, 2wal 1441 . . 3  wff  A. w
( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
) )
2524, 1wal 1441 . 2  wff  A. z A. w ( ( z  e.  w  /\  w  e.  x )  ->  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
) )
2625, 13wex 1662 1  wff  E. y A. z A. w ( ( z  e.  w  /\  w  e.  x
)  ->  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
) )
Colors of variables: wff setvar class
This axiom is referenced by:  zfac  8887  ac2  8888
  Copyright terms: Public domain W3C validator