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Theorem ac2 8082
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 8083 is easier to understand.) Note: aceq0 7740 shows the logical equivalence to ax-ac 8080. (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
Dummy variable  t is distinct from all other variables.

Proof of Theorem ac2
StepHypRef Expression
1 ax-ac 8080 . 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 7740 . 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 202 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 6    <-> wb 178    /\ wa 360   A.wal 1528   E.wex 1529    = wceq 1624    e. wcel 1685   A.wral 2544   E.wrex 2545   E!wreu 2546
This theorem is referenced by:  ac3  8083
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-ac 8080
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ral 2549  df-rex 2550  df-reu 2551
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