(* Title: ZF/AC/WO_AC.ML
ID: $Id$
Author: Krzysztof Grabczewski
Lemmas used in the proofs like WO? ==> AC?
*)
open WO_AC;
Goal "[| well_ord(Union(A),r); 0~:A; B:A |] \
\ ==> (THE b. first(b,B,r)) : B";
by (fast_tac (claset() addSEs [well_ord_imp_ex1_first RS theI RS
(first_def RS def_imp_iff RS iffD1 RS conjunct1)]) 1);
qed "first_in_B";
Goal "[| well_ord(Union(A), R); 0~:A |] ==> EX f. f:(PROD X:A. X)";
by (fast_tac (claset() addSEs [first_in_B] addSIs [lam_type]) 1);
qed "ex_choice_fun";
Goal "well_ord(A, R) ==> EX f. f:(PROD X: Pow(A)-{0}. X)";
by (fast_tac (claset() addSEs [well_ord_subset RS ex_choice_fun]) 1);
qed "ex_choice_fun_Pow";