| Entry |
Value |
|
Name |
EXISTS_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!t1 t2 t3. (t1 /\ t2) /\ t3 <=> t1 /\ t2 /\ t3!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 |
