| Entry |
Value |
|
Name |
UNIV_NOT_EMPTY |
|
Conclusion |
~(UNIV = {}) |
|
Constructive Proof |
No |
|
Axiom |
!t. t \/ ~t
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d) |
|
Classical Lemmas |
!t. ~ ~t <=> t!t. (t <=> T) \/ (t <=> F)!p. ~(?x. p x) <=> (!x. ~p x)!s t. ~(s = t) <=> (?x. x IN t <=> ~(x IN s))!s. (?x. x IN s) <=> ~(s = {}) |
|
Constructive Lemmas |
T!x y. x = y <=> y = x!x y. x = y ==> y = x!x. ~(x IN {})!x. x = x!x. x IN UNIV!t. (!x. t) <=> t!t. (?x. 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. F \/ t <=> t!t. T \/ t <=> T!t. t \/ F <=> t!t. t \/ T <=> T!t. t \/ t <=> t!t. t ==> t!t. (t <=> T) \/ (t <=> F)!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!p. ~(?x. p x) <=> (!x. ~p x)!s t. (!x. x IN s <=> x IN t) <=> s = t!s t. ~(s = t) <=> (?x. x IN t <=> ~(x IN s))!s t. (!x. x IN s <=> x IN t) ==> s = t!s. (?x. x IN s) <=> ~(s = {})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 q. !r. (p ==> r) ==> (q ==> r) ==> r)(!) = (\p. p = (\x. T))(?) = (\p. !q. (!x. p x ==> q) ==> q){} = {x | F}UNIV = {x | T} |
|
Contained Package |
set-thm |
|
Comment |
Standard HOL library retrieved from OpenTheory |
