Entry Value
Name INSERT_POWERSET
Conclusion !a t. {s | s SUBSET a INSERT t} = {s | s SUBSET t} UNION IMAGE (\s. a INSERT s) {s | s SUBSET t}
Constructive Proof No
Axiom 
!t. t \/ ~t 
(\a. a = (\b. (\c. c) = (\c. c))) (\d. (\e. d e) = d) 
(\a. a = (\b. (\c. c) = (\c. c)))
(\d. (\e. e = (\f. (\c. c) = (\c. c)))
     (\g. (\h i.
               (\j k.
                    (\l. l j k) =
                    (\m. m ((\c. c) = (\c. c)) ((\c. c) = (\c. c))))
               h
               i <=>
               h)
          (d g)
          (d ((@) d))))
Classical Lemmas
  • !x y s. (x INSERT s) DELETE y = (if x = y then s DELETE y else x INSERT (s DELETE y))
  • !x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t
  • !x s. s DELETE x = s <=> ~(x IN s)
  • !x s. DISJOINT s {x} <=> ~(x IN s)
  • !x s. DISJOINT {x} s <=> ~(x IN s)
  • !x s. x IN s ==> x INSERT (s DELETE x) = s
  • !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2
  • !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2
  • !t. ~ ~t <=> t
  • !t. (t <=> T) \/ (t <=> F)
  • !p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)
  • !s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u
  • !s t. s DIFF t = s <=> DISJOINT s t
    		•
    Constructive Lemmas
  • T
  • !x y s. x IN y INSERT s <=> x = y \/ x IN s
  • !x y s. (x INSERT s) DELETE y = (if x = y then s DELETE y else x INSERT (s DELETE y))
  • !t1 t2. (if F then t1 else t2) = t2
  • !t1 t2. (if T then t1 else t2) = t1
  • !x y. x = y <=> y = x
  • !x y. x IN {y} <=> x = y
  • !x y. x = y ==> y = x
  • !x s t. s SUBSET x INSERT t <=> s DELETE x SUBSET t
  • !x s t. x INSERT s UNION t = x INSERT (s UNION t)
  • !x s. s DELETE x = s <=> ~(x IN s)
  • !x s. DISJOINT s {x} <=> ~(x IN s)
  • !x s. DISJOINT {x} s <=> ~(x IN s)
  • !x s. s DIFF {x} = s DELETE x
  • !x s. x INSERT s = {y | y = x \/ y IN s}
  • !x s. {x} UNION s = x INSERT s
  • !x s. x IN s ==> x INSERT (s DELETE x) = s
  • !x s. s DELETE x SUBSET s
  • !x. ~(x IN {})
  • !x. x = x
  • !x. (@y. y = x) = x
  • !y s f. y IN IMAGE f s <=> (?x. y = f x /\ x IN s)
  • !t1 t2 t3. (t1 \/ t2) \/ t3 <=> t1 \/ t2 \/ t3
  • !t1 t2. ~(t1 /\ t2) <=> ~t1 \/ ~t2
  • !t1 t2. ~(t1 \/ t2) <=> ~t1 /\ ~t2
  • !t1 t2. t1 /\ t2 <=> t2 /\ t1
  • !t1 t2. t1 \/ t2 <=> t2 \/ t1
  • !a b. (a <=> b) ==> a ==> b
  • !t. (!x. t) <=> t
  • !t. ~ ~t <=> t
  • !t. F /\ t <=> F
  • !t. T /\ t <=> t
  • !t. t /\ F <=> F
  • !t. t /\ T <=> 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
  • !f s. IMAGE f s = {y | ?x. x IN s /\ y = f x}
  • !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 ==> p ((@) p)
  • !p c x y. p (if c then x else y) <=> (c ==> p x) /\ (~c ==> p y)
  • !p f q. (!z. z IN {f x | p x} ==> q z) <=> (!x. p x ==> q (f x))
  • !p f q. (!z. z IN {f x y | p x y} ==> q z) <=> (!x y. p x y ==> q (f x y))
  • !p f q. (!z. z IN {f w x y | p w x y} ==> q z) <=> (!w x y. p w x y ==> q (f w x y))
  • !p f s. (!y. y IN IMAGE f s ==> p y) <=> (!x. x IN s ==> p (f x))
  • !s x y. x IN s DELETE y <=> x IN s /\ ~(x = y)
  • !s x. s DELETE x = {y | y IN s /\ ~(y = x)}
  • !s t x. x IN s DIFF t <=> x IN s /\ ~(x IN t)
  • !s t x. x IN s INTER t <=> x IN s /\ x IN t
  • !s t x. x IN s UNION t <=> x IN s \/ x IN t
  • !s t u. s SUBSET t UNION u <=> s DIFF t SUBSET u
  • !s t u. s UNION t SUBSET u <=> s SUBSET u /\ t SUBSET u
  • !s t u. s SUBSET t /\ t SUBSET u ==> s SUBSET u
  • !s t. (!x. x IN s <=> x IN t) <=> s = t
  • !s t. s DIFF t = s <=> DISJOINT s t
  • !s t. DISJOINT s t <=> s INTER t = {}
  • !s t. DISJOINT s t <=> DISJOINT t s
  • !s t. s SUBSET t <=> (!x. x IN s ==> x IN t)
  • !s t. s DIFF t = {x | x IN s /\ ~(x IN t)}
  • !s t. s INTER t = {x | x IN s /\ x IN t}
  • !s t. s INTER t = t INTER s
  • !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 t. s SUBSET t /\ t SUBSET s ==> s = t
  • !s t. s DIFF t SUBSET s
  • !s. {} UNION s = s
  • !s. s UNION {} = s
  • F <=> (!p. p)
  • T <=> (\p. p) = (\p. p)
  • ~F <=> T
  • ~T <=> F
  • (~) = (\p. p ==> F)
  • COND = (\t t1 t2. @x. ((t <=> T) ==> x = t1) /\ ((t <=> F) ==> x = t2))
  • (/\) = (\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)
  • (?) = (\p. p ((@) p))
  • {} = {x | F}
    		•
    Contained Package set-thm
    Comment Standard HOL library retrieved from OpenTheory
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