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Axiom ax-ac 8082
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The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry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 8085 for a more detailed explanation. Theorem ac2 8084 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 8089 is slightly shorter when the biconditional of ax-ac 8082 is expanded into implication and negation. In axac3 8087 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8292 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8116, ac5 8101, and ac7 8097. The Axiom of Regularity ax-reg 7303 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7754. Equivalents to AC are the well-ordering theorem weth 8119 and Zorn's lemma zorn 8131. See ac4 8099 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7303 for derivation of AC equivalents, we provide ax-ac2 8086 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8086 from ax-ac 8082 is shown by theorem axac2 8090, and the reverse derivation by axac 8091. Therefore, new proofs should normally use ax-ac2 8086 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 1688 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1688 . . . . . 6  wff  w  e.  x
63, 5wa 360 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  set  u
87, 2wel 1688 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  set  t
102, 9wel 1688 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 360 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1688 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  set  y
149, 13wel 1688 . . . . . . . . . . 11  wff  t  e.  y
1512, 14wa 360 . . . . . . . . . 10  wff  ( u  e.  t  /\  t  e.  y )
1611, 15wa 360 . . . . . . . . 9  wff  ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y ) )
1716, 9wex 1530 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  set  v
197, 18weq 1626 . . . . . . . 8  wff  u  =  v
2017, 19wb 178 . . . . . . 7  wff  ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
2120, 7wal 1529 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1530 . . . . 5  wff  E. v A. u ( E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
236, 22wi 6 . . . 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 1529 . . 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 1529 . 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 1530 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  8083  ac2  8084
  Copyright terms: Public domain W3C validator