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Theorem ac3 8088
Description: Axiom of Choice using abbreviations. The logical equivalence to ax-ac 8085 can be established by chaining aceq0 7745 and aceq2 7746. A standard textbook version of AC is derived from this one in dfac2a 7756, and this version of AC is derived from the textbook version in dfac2 7757.

The following sketch will help you understand this version of the axiom. Given any set  x, the axiom says that 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. Using the Axiom of Regularity, we can show that  y is really a set of ordered pairs, very similar to the ordered pair construction opthreg 7319. The key theorem for this (used in the proof of dfac2 7757) is preleq 7318. With this modified definition of ordered pair, it can be seen that  y is actually a choice function on the members of  x.

For example, suppose  x  =  { { 1 ,  2 } ,  { 1 ,  3 } ,  { 2 ,  3 ,  4 } }. Let us try  y  =  { { { 1 ,  2 } ,  1 } ,  { { 1 ,  3 } , 
1 } ,  { { 2 ,  3 ,  4 } ,  2 } }. For the member (of  x)  z  =  {
1 ,  2 }, the only assignment to  w and  v that satisfies the axiom is  w  =  1 and  v  =  { { 1 ,  2 } , 
1 }, so there is exactly one  w as required. We verify the other two members of  x similarly. Thus  y satisfies the axiom. Using our modified ordered pair definition, we can say that  y corresponds to the choice function  { <. { 1 ,  2 } , 
1 >. ,  <. { 1 ,  3 } , 
1 >. ,  <. { 2 ,  3 ,  4 } ,  2 >. }. Of course other choices for  y will also satisfy the axiom, for example  y  =  { { { 1 ,  2 } ,  2 } ,  { { 1 ,  3 } , 
1 } ,  { { 2 ,  3 ,  4 } ,  4 } }. What AC tells us is that there exists at least one such  y, but it doesn't tell us which one.

(New usage is discouraged.) (Contributed by NM, 19-Jul-1996.)

Assertion
Ref Expression
ac3  |-  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
z  e.  v  /\  w  e.  v )
)
Distinct variable group:    x, y, z, w, v

Proof of Theorem ac3
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 ac2 8087 . 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
)
2 aceq2 7746 . 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  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
z  e.  v  /\  w  e.  v )
) )
31, 2mpbi 199 1  |-  E. y A. z  e.  x  ( z  =/=  (/)  ->  E! w  e.  z  E. v  e.  y  (
z  e.  v  /\  w  e.  v )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   E!wreu 2545   (/)c0 3455
This theorem is referenced by:  axac3OLD  8091  axac2  8093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-ac 8085
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-v 2790  df-dif 3155  df-nul 3456
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