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(* Title: ZF/AC/AC1_WO2.ML


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ID: $Id$


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Author: Krzysztof Gr`abczewski


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The proof of AC1 ==> WO2


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*)


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open AC1_WO2;


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val [prem] = goal thy "f : (PROD X:Pow(x)  {0}. X) ==> \


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\ ?g(f) : bij(x, LEAST i. HH(lam X:Pow(x){0}. {f`X}, x, i) = {x})";


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by (resolve_tac [bij_Least_HH_x RS bij_converse_bij] 1);


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by (resolve_tac [f_subsets_imp_UN_HH_eq_x] 1);


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by (resolve_tac [lam_type RS apply_type] 1 THEN (assume_tac 2));


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by (fast_tac (AC_cs addSDs [equals0D, prem RS apply_type]) 1);


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by (fast_tac (AC_cs addSIs [prem RS Pi_weaken_type]) 1);


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val lemma1 = uresult();


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goalw thy [AC1_def, WO2_def, eqpoll_def] "!!Z. AC1 ==> WO2";


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by (resolve_tac [allI] 1);


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by (eres_inst_tac [("x","Pow(A){0}")] allE 1);


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by (fast_tac (AC_cs addSDs [lemma1] addSIs [Ord_Least]) 1);

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qed "AC1_WO2";
