Entry Value
Name INSERT_UNIV
Conclusion !x. x INSERT UNIV = UNIV
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 s. x IN y INSERT s <=> x = y \/ x IN s
  • !x y. x = y <=> y = x
  • !x y. x = y ==> y = x
  • !x s t. x INSERT s UNION t = x INSERT (s UNION t)
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x s. {x} UNION s = x INSERT s
  • !x. ~(x IN {})
  • !x. x = x
  • !x. x IN UNIV
  • !t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3
  • !t1 t2. t1 \/ t2 <=> t2 \/ t1
  • !t. (!x. 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
  • !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 UNION t <=> x IN s \/ x IN t
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. s UNION t = {x | x IN s \/ x IN t}
  • !s t. s UNION t = t UNION s
  • !s t. (!x. x IN s <=> x IN t) ==> s = t
  • !s. {} UNION s = s
  • !s. UNIV UNION s = UNIV
  • !s. s UNION {} = s
  • !s. s UNION UNIV = UNIV
  • 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
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