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Axiom ax-ac 8101
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The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry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 8104 for a more detailed explanation. Theorem ac2 8103 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 8108 is slightly shorter when the biconditional of ax-ac 8101 is expanded into implication and negation. In axac3 8106 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8311 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8135, ac5 8120, and ac7 8116. The Axiom of Regularity ax-reg 7322 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7773. Equivalents to AC are the well-ordering theorem weth 8138 and Zorn's lemma zorn 8150. See ac4 8118 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7322 for derivation of AC equivalents, we provide ax-ac2 8105 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8105 from ax-ac 8101 is shown by theorem axac2 8109, and the reverse derivation by axac 8110. Therefore, new proofs should normally use ax-ac2 8105 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  set  z
2 vw . . . . . . 7  set  w
31, 2wel 1697 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1697 . . . . . 6  wff  w  e.  x
63, 5wa 358 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  set  u
87, 2wel 1697 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  set  t
102, 9wel 1697 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 358 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1697 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  set  y
149, 13wel 1697 . . . . . . . . . . 11  wff  t  e.  y
1512, 14wa 358 . . . . . . . . . 10  wff  ( u  e.  t  /\  t  e.  y )
1611, 15wa 358 . . . . . . . . 9  wff  ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y ) )
1716, 9wex 1531 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  set  v
197, 18weq 1633 . . . . . . . 8  wff  u  =  v
2017, 19wb 176 . . . . . . 7  wff  ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
2120, 7wal 1530 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1531 . . . . 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 1530 . . 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 1530 . 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 1531 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 set class
This axiom is referenced by:  zfac  8102  ac2  8103
  Copyright terms: Public domain W3C validator