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Theorem ac8 8121
Description: An Axiom of Choice equivalent. Given a family  x of mutually disjoint nonempty sets, there exists a set  y containing exactly one member from each set in the family. Theorem 6M(4) of [Enderton] p. 151. (Contributed by NM, 14-May-2004.)
Assertion
Ref Expression
ac8  |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y
) )
Distinct variable group:    x, z, y, w, v

Proof of Theorem ac8
StepHypRef Expression
1 dfac5 7757 . 2  |-  (CHOICE  <->  A. x
( ( A. z  e.  x  z  =/=  (/) 
/\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w
)  =  (/) ) )  ->  E. y A. z  e.  x  E! v 
v  e.  ( z  i^i  y ) ) )
21axaci 8097 1  |-  ( ( A. z  e.  x  z  =/=  (/)  /\  A. z  e.  x  A. w  e.  x  ( z  =/=  w  ->  ( z  i^i  w )  =  (/) ) )  ->  E. y A. z  e.  x  E! v  v  e.  ( z  i^i  y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1530    = wceq 1625    e. wcel 1686   E!weu 2145    =/= wne 2448   A.wral 2545    i^i cin 3153   (/)c0 3457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-ac2 8091
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ac 7745
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