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Axiom ax-ac 8328
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The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry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 8331 for a more detailed explanation. Theorem ac2 8330 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 8334 is slightly shorter when the biconditional of ax-ac 8328 is expanded into implication and negation. In axac3 8333 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8537 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8361, ac5 8346, and ac7 8342. The Axiom of Regularity ax-reg 7549 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 8000. Equivalents to AC are the well-ordering theorem weth 8364 and Zorn's lemma zorn 8376. See ac4 8344 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7549 for derivation of AC equivalents, we provide ax-ac2 8332 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8332 from ax-ac 8328 is shown by theorem axac2 8335, and the reverse derivation by axac 8336. Therefore, new proofs should normally use ax-ac2 8332 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 1726 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1726 . . . . . 6  wff  w  e.  x
63, 5wa 359 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  set  u
87, 2wel 1726 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  set  t
102, 9wel 1726 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 359 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1726 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  set  y
149, 13wel 1726 . . . . . . . . . . 11  wff  t  e.  y
1512, 14wa 359 . . . . . . . . . 10  wff  ( u  e.  t  /\  t  e.  y )
1611, 15wa 359 . . . . . . . . 9  wff  ( ( u  e.  w  /\  w  e.  t )  /\  ( u  e.  t  /\  t  e.  y ) )
1716, 9wex 1550 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  set  v
197, 18weq 1653 . . . . . . . 8  wff  u  =  v
2017, 19wb 177 . . . . . . 7  wff  ( E. t ( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)  <->  u  =  v
)
2120, 7wal 1549 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1550 . . . . 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 1549 . . 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 1549 . 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 1550 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  8329  ac2  8330
  Copyright terms: Public domain W3C validator