equal
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183 by (etac Ord_ordertype 2); |
183 by (etac Ord_ordertype 2); |
184 by (rtac exI 1); |
184 by (rtac exI 1); |
185 by (etac (ordermap_bij RS bij_converse_bij) 1); |
185 by (etac (ordermap_bij RS bij_converse_bij) 1); |
186 qed "well_ord_cardinal_eqpoll"; |
186 qed "well_ord_cardinal_eqpoll"; |
187 |
187 |
188 val Ord_cardinal_eqpoll = well_ord_Memrel RS well_ord_cardinal_eqpoll |
188 bind_thm ("Ord_cardinal_eqpoll", well_ord_Memrel RS well_ord_cardinal_eqpoll); |
189 |> standard; |
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190 |
189 |
191 goal Cardinal.thy |
190 goal Cardinal.thy |
192 "!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y"; |
191 "!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y"; |
193 by (rtac (eqpoll_sym RS eqpoll_trans) 1); |
192 by (rtac (eqpoll_sym RS eqpoll_trans) 1); |
194 by (etac well_ord_cardinal_eqpoll 1); |
193 by (etac well_ord_cardinal_eqpoll 1); |
339 by (fast_tac (ZF_cs addSIs [nat_0_le]) 1); |
338 by (fast_tac (ZF_cs addSIs [nat_0_le]) 1); |
340 by (rtac ballI 1); |
339 by (rtac ballI 1); |
341 by (eres_inst_tac [("n","n")] natE 1); |
340 by (eres_inst_tac [("n","n")] natE 1); |
342 by (asm_simp_tac (ZF_ss addsimps [inj_def, succI1 RS Pi_empty2]) 1); |
341 by (asm_simp_tac (ZF_ss addsimps [inj_def, succI1 RS Pi_empty2]) 1); |
343 by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [inj_succ_succD]) 1); |
342 by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [inj_succ_succD]) 1); |
344 qed "nat_lepoll_imp_le_lemma"; |
343 val nat_lepoll_imp_le_lemma = result(); |
345 val nat_lepoll_imp_le = nat_lepoll_imp_le_lemma RS bspec RS mp |> standard; |
344 |
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345 bind_thm ("nat_lepoll_imp_le", nat_lepoll_imp_le_lemma RS bspec RS mp); |
346 |
346 |
347 goal Cardinal.thy |
347 goal Cardinal.thy |
348 "!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n"; |
348 "!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n"; |
349 by (rtac iffI 1); |
349 by (rtac iffI 1); |
350 by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |
350 by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |