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

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Example text

Hence, a p p l y - 55 AXIOMATIC SET THEORY PROOF OF ( i x ) . 10 By ( v i i i ) , we have B n DF-' ( B n D F - ' ) = B n DF-'. 10 ( F - ' * % 8 ) u (F-'* x D 0 F - l ) = F - l * % B . (xi, = F*A f o r some A . e. ( x i ) and f x i i ) a r e ' l e f t t o t h e r e a d e r . We a l s o have d i s t r i b u t i v i t y o f i n v e r s e image o f f u n c t i o n s w i t h generalized intersection. 9 THEOREM SCHEMA, L& r be a t m and 6 a 6o/un&. On t h e o t h e r hand, i f x \$ implies that f o r a l l such r .

A i s a b c t ad codecl dotl A in [ R , A I . R i s not enough t o determine A ; because A might be d i f f e r e n t from D R. W i t h j u s t R we could not have superclasse A with A ( O ) , because f o r a l l x E D R, R*{x} f 0. 3 THEOREM SCHEMA, L e t F be a u m q aperraLLiun. Then, Wx(xEB + F ( x ) = 0 ) + [ F (x) : x E A ] = [F(x) : x e A U B ] AXIOMATIC S E T T H E O R Y PROOF, Suppose t h a t F ( x ) = 0, f o r X E 8. 41 We have, [F(x) : x g A u B ] = u (F(x) x ( x } :x E A u B ) = B u t , by hypothesis, Therefore, : x E B ) = 0.

3. PROOF, (RoS) THEOREM, Assume R R2 -R C A S2 C S A R - 2 C _ R A S2 C -S A R o S oS = S OR = SoR. e. + (R oS)2 C - R oS. Then, 2 = R o ( S o R ) o S = ( R o R ) o ( S o S ) = R 2 O S2 R i s den6e, i f R C R -+ 5 R o S . R i s an 3 z(xRz A z R y ) ) . e. i f R = R - l and 2 R C R. There a r e s e v e r a l o t h e r ways o f s a y i n g t h a t R i s an equivalence relation: (i)says t h a t i f R i s symmetric and t r a n s i t i v e , then i t i s r e f l e x i v e . N o t i c e t h a t t h i s i s n o t t r u e f o r r e f l e x i v e i n A.