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
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