| Entry |
Value |
|
Name |
FORALL_IN_UNIONS |
|
Conclusion |
!p s. (!x. x IN UNIONS s ==> p x) <=> (!t x. t IN s /\ x IN t ==> p x) |
|
Constructive Proof |
Yes |
|
Axiom |
N|A |
|
Classical Lemmas |
N|A |
|
Constructive Lemmas |
T!x. x = x!t. (!x. t) <=> t!t. (F <=> t) <=> ~t!t. (T <=> t) <=> t!t. (t <=> F) <=> ~t!t. (t <=> T) <=> t!f y. (\x. f x) y = f y!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 q. (?x. p x) ==> q <=> (!x. p x ==> q)!p. (!x y. p x y) <=> (!y x. p x y)!s x. x IN UNIONS s <=> (?t. t IN s /\ x IN t)!s. UNIONS s = {x | ?u. u IN s /\ x IN u}F <=> (!p. p)T <=> (\p. p) = (\p. p)(~) = (\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) |
|
Contained Package |
set-thm |
|
Comment |
Standard HOL library retrieved from OpenTheory |
