MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ac2 Unicode version

Theorem ac2 8103
Description: Axiom of Choice equivalent. By using restricted quantifiers, we can express the Axiom of Choice with a single explicit conjunction. (If you want to figure it out, the rewritten equivalent ac3 8104 is easier to understand.) Note: aceq0 7761 shows the logical equivalence to ax-ac 8101. (New usage is discouraged.) (Contributed by NM, 18-Jul-1996.)
Assertion
Ref Expression
ac2  |-  E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y 
( z  e.  u  /\  v  e.  u
)
Distinct variable group:    x, y, z, w, v, u

Proof of Theorem ac2
Dummy variable  t is distinct from all other variables.
StepHypRef Expression
1 ax-ac 8101 . 2  |-  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
) )
2 aceq0 7761 . 2  |-  ( E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y  ( z  e.  u  /\  v  e.  u )  <->  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
) ) )
31, 2mpbir 200 1  |-  E. y A. z  e.  x  A. w  e.  z  E! v  e.  z  E. u  e.  y 
( z  e.  u  /\  v  e.  u
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   E!wreu 2558
This theorem is referenced by:  ac3  8104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-ac 8101
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-reu 2563
  Copyright terms: Public domain W3C validator