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Axiom ax-ac 8295
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The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry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 8298 for a more detailed explanation. Theorem ac2 8297 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 8301 is slightly shorter when the biconditional of ax-ac 8295 is expanded into implication and negation. In axac3 8300 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8504 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8328, ac5 8313, and ac7 8309. The Axiom of Regularity ax-reg 7516 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7967. Equivalents to AC are the well-ordering theorem weth 8331 and Zorn's lemma zorn 8343. See ac4 8311 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7516 for derivation of AC equivalents, we provide ax-ac2 8299 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8299 from ax-ac 8295 is shown by theorem axac2 8302, and the reverse derivation by axac 8303. Therefore, new proofs should normally use ax-ac2 8299 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 1722 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1722 . . . . . 6  wff  w  e.  x
63, 5wa 359 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  set  u
87, 2wel 1722 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  set  t
102, 9wel 1722 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 359 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1722 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  set  y
149, 13wel 1722 . . . . . . . . . . 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 1547 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  set  v
197, 18weq 1650 . . . . . . . 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 1546 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1547 . . . . 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 1546 . . 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 1546 . 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 1547 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  8296  ac2  8297
  Copyright terms: Public domain W3C validator