EMPTY_DISJOINT

Entry	Value
Name	EMPTY_DISJOINT
Conclusion	!s. DISJOINT {} s
Constructive Proof	Yes
Axiom	(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d)
Classical Lemmas	N\|A
Constructive Lemmas	T !x y. x = y <=> y = x !x y. x = y ==> y = x !x. ~(x IN {}) !x. x = x !t1 t2. t1 /\ t2 <=> t2 /\ t1 !t. (!x. t) <=> t !t. F /\ t <=> F !t. T /\ t <=> t !t. t /\ F <=> F !t. t /\ T <=> t !t. t /\ t <=> t !t. (F <=> t) <=> ~t !t. (T <=> t) <=> t !t. (t <=> F) <=> ~t !t. (t <=> T) <=> t !t. F ==> t <=> T !t. T ==> t <=> t !t. t ==> F <=> ~t !t. t ==> T <=> T !t. t ==> t !f y. (\x. f x) y = f y !f g. (!x. f x = g x) <=> f = g !f g. (!x. f x = g x) ==> f = g !t. (\x. t x) = t !p a. (?x. a = x /\ p x) <=> p a !p x. x IN GSPEC p <=> p x !p x. x IN {y \| p y} <=> p x !s t x. x IN s INTER t <=> x IN s /\ x IN t !s t. (!x. x IN s <=> x IN t) <=> s = t !s t. DISJOINT s t <=> s INTER t = {} !s t. s INTER t = {x \| x IN s /\ x IN t} !s t. s INTER t = t INTER s !s t. (!x. x IN s <=> x IN t) ==> s = t !s. {} INTER s = {} !s. s INTER {} = {} F <=> (!p. p) T <=> (\p. p) = (\p. p) ~F <=> T ~T <=> F (~) = (\p. p ==> F) (/\) = (\p q. (\f. f p q) = (\f. f T T)) (==>) = (\p q. p /\ q <=> p) (!) = (\p. p = (\x. T)) (?) = (\p. !q. (!x. p x ==> q) ==> q) {} = {x \| F}
Contained Package	set-thm
Comment	Standard HOL library retrieved from OpenTheory