let Y, X, Z be set ; :: thesis: ( Y c= Tarski-Class X & Z c= Tarski-Class X implies [:Y,Z:] c= Tarski-Class X )
assume A1:
( Y c= Tarski-Class X & Z c= Tarski-Class X )
; :: thesis: [:Y,Z:] c= Tarski-Class X
let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in [:Y,Z:] or x in Tarski-Class X )
assume A2:
x in [:Y,Z:]
; :: thesis: x in Tarski-Class X
then A3:
x = [(x `1 ),(x `2 )]
by MCART_1:23;
( x `1 in Y & x `2 in Z )
by A2, MCART_1:10;
hence
x in Tarski-Class X
by A1, A3, Th31; :: thesis: verum