MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-ac Unicode version

Axiom ax-ac 8085
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The unpublished version given here says that given any set  x, there exists a  y that is a collection of unordered pairs, one pair for each non-empty member of  x. One entry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 8088 for a more detailed explanation. Theorem ac2 8087 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 8092 is slightly shorter when the biconditional of ax-ac 8085 is expanded into implication and negation. In axac3 8090 we allow the constant CHOICE to represent the Axiom of Choice; this simplifies the representation of theorems like gchac 8295 (the Generalized Continuum Hypothesis implies the Axiom of Choice).

Standard textbook versions of AC are derived as ac8 8119, ac5 8104, and ac7 8100. The Axiom of Regularity ax-reg 7306 (among others) is used to derive our version from the standard ones; this reverse derivation is shown as theorem dfac2 7757. Equivalents to AC are the well-ordering theorem weth 8122 and Zorn's lemma zorn 8134. See ac4 8102 for comments about stronger versions of AC.

In order to avoid uses of ax-reg 7306 for derivation of AC equivalents, we provide ax-ac2 8089 (due to Kurt Maes), which is equivalent to the standard AC of textbooks. The derivation of ax-ac2 8089 from ax-ac 8085 is shown by theorem axac2 8093, and the reverse derivation by axac 8094. Therefore, new proofs should normally use ax-ac2 8089 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 1685 . . . . . 6  wff  z  e.  w
4 vx . . . . . . 7  set  x
52, 4wel 1685 . . . . . 6  wff  w  e.  x
63, 5wa 358 . . . . 5  wff  ( z  e.  w  /\  w  e.  x )
7 vu . . . . . . . . . . . 12  set  u
87, 2wel 1685 . . . . . . . . . . 11  wff  u  e.  w
9 vt . . . . . . . . . . . 12  set  t
102, 9wel 1685 . . . . . . . . . . 11  wff  w  e.  t
118, 10wa 358 . . . . . . . . . 10  wff  ( u  e.  w  /\  w  e.  t )
127, 9wel 1685 . . . . . . . . . . 11  wff  u  e.  t
13 vy . . . . . . . . . . . 12  set  y
149, 13wel 1685 . . . . . . . . . . 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 1528 . . . . . . . 8  wff  E. t
( ( u  e.  w  /\  w  e.  t )  /\  (
u  e.  t  /\  t  e.  y )
)
18 vv . . . . . . . . 9  set  v
197, 18weq 1624 . . . . . . . 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 1527 . . . . . 6  wff  A. u
( E. t ( ( u  e.  w  /\  w  e.  t
)  /\  ( u  e.  t  /\  t  e.  y ) )  <->  u  =  v )
2221, 18wex 1528 . . . . 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 1527 . . 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 1527 . 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 1528 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  8086  ac2  8087
  Copyright terms: Public domain W3C validator