# Axiomatic Set Theory: Impredicative Theories of Classes by R. Chuaqui By R. Chuaqui

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E. y Therefore FIX E DEFINITION BA i s t h e & a n 06 F'x = :\$I, t h e n f o r a l l Xo... f o r some y nx 0.. x n-1 :\$I. e. Xn-l, and hence F'xE r x c ~ - l *n '0' * 'Xn-1 I 6uncLionn w L t h domain 8 and m n g e included i n A. e. t h a t t h e r e is a w i l l n o t be much used i n P a r t 2. On t h e o t h e r hand A ( ? ) i s t h e no;tion t h a t says t h a t f 12 a 6unOtiun w a h domain 8 m d tange i n c h d e d in A. W i t h t h i s n o t i o n , we can express t h a t ? i s a f u n c t i o n by D F V ( F ) .

F i r s t , t h e theorem f o r 0, 2-monotone o p e r a t i o n s . THEOREM SCHEMA, L e t F be a unmq opehation. 4 W A W B ( A-c B c Z + F ( A )CF(B)LZ)+ n {X : X c - Z A F(X) = X ] A U = LJ Then ~C~U(F(C)=CAF(U)=DAC= { X :X C - Z A F(X) = A } ) . C i s t h e l e a s t f i x e d p o i n t and Q i s t h e g r e a t e s t f i x e d p o i n t . Thus, t h e c o n c l u s i o n can a l s o be w r i t t e n , WA W B ( A - c B-c Z + F ( A ) & F ( B ) 5 Z ) + F ( n { X : X-C Z A F ( X )= X } ) = = n {x : x A F(X) = c - z A F ( x ) = X I A F( u { x : x c - z A F(x)= XI.

As f o l - But 0 a, b E V Cu . U) . Therefore, by Ax Sub 0 E Y.. ( { a , b ) E V A 3 U(U u c -u A a u)). 3 LEMMA ( B ) PROOF, Suppose t h a t a, b E V and a + 6 ( t h e p r o o f f o r t h e c a s e L e t A = { a , b } . We have, -* E a = b i s similar). 3x 3 y ( x # y A x , y E A ) . Using Ax Ref we o b t a i n , 3u( Uu c -u A 3x 3y (x,y 5u A x # y A x,y E A n u ) ) . Hence, t h e r e i s a t r a n s i t i v e s e t u, such t h a t , (1) 3 x 3 y ( x # y A x,y E A n u). Therefore, s i n c e A n u -A C = ( a , b } , A n u = 0, A n u = { a } , An u = (6) 32 ROLAND0 C H U A Q U I o r Anu = { a , b } .