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

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E . ( y , z ) \$ R . suchthat(x,z)ERand(z,y)€ The p r o o f Then (x,y )E of R Then t h e r e i s a z Thus, x + y . 6 REMARKS, I t i s i n t e r e s t i n g t o n o t e t h a t u s i n g some o f t h e s e i d e n t i t i e s , i t i s p o s s i b l e t o r e p l a c e e v e r y Boolean ( p r o p o s i t i o n a l c a l c u l u s ) combination o f e q u a t i o n s and i n e q u a l i t i e s by one equation. For i n s t a n c e , take (1) R = S V R ' = S' s t a r t with, l( R = S V R' = S ' ) . T h i s i s e q u i v a l e n t t o f?

Then W X W Y ( ( X-c Y c-A - + F ( X ) cF(Y))A(XcY5B+G(X) cG(Y)))A F(A) c 8 A G(B) L A A nA = 0 = 1 2 B 1 nB 2 + 3 A1 3 A 2 3B1 3B2 (A = A U A A 1 2 A F ( A 2 ) = B1 A G ( B 2 ) = A1) . B = B1UB2 A L e t F and G be monotone o p e r a t i o n s f o r s u b c l a s s e s o f A and PROOF, r e s p e c t i v e l y and suppose F ( A ) 2 B and G ( B ) 5 A. 5, there a r e A1, B1 such t h a t A1 E G ( B ) , B1 c F ( A ) , F ( A % A 1 ) = B1, and G ( B % B l ) =A1. B, L e t A2 = A % A 1 and B2 = 8%Bl. We have, A1 L G ( B ) 5A t h u s A1 C A and B1 5 B.

R ~ R -=I R O A R~ = R A R o ( v x v ) oR c RUR-~. LO(R) The proof i s l e f t t o the reader. F i n a l l y , we study well-founded r e l a t i o n s . 19 (i) DEF IN IT ION, - oR(x)= ( i i ) WF(R) (R-'*{X>) % 1x1 . W A(A c -DRu D R - l A A f 0 -+ 3 x ( x e A A A n OR(x)=O)). e. OR(x) = ( q : y + x A yRx1. 20 ROLAND0 CHUAQUI VA(A C - D R U DU-' A A # 0 3 x(xEA A Wy(yEA A qRx+y=x))). + THEOREM, ( i ) and ( i i ) s i m p l i f y t h e d e f i n i t i o n o f well-foundedness f o r r e f l e x i v e ( i i i ) asserts t h a t i f R i s welland i r r e f l e x i v e r e l a t i o n s , r e s p e c t i v e l y .