Nuprl Lemma : free-from-atom-pair-iff
∀[a:Atom1]. ∀[X:Type]. ∀[Y:𝕌']. ∀[x:X]. ∀[y:Y]. uiff(a#<x, y>:x:X × Y;a#x:X ∧ a#y:Y)
Proof
Definitions occuring in Statement :
free-from-atom: a#x:T,
atom: Atom$n,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
prop: ℙ
Lemmas referenced :
free-from-atom-pair-components,
free-from-atom_wf,
free-from-atom-pair,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
hypothesis,
productElimination,
because_Cache,
independent_pairEquality,
freeFromAtomAxiom,
cumulativity,
instantiate,
productEquality,
dependent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
atomnEquality
Latex:
\mforall{}[a:Atom1]. \mforall{}[X:Type]. \mforall{}[Y:\mBbbU{}']. \mforall{}[x:X]. \mforall{}[y:Y]. uiff(a\#<x, y>:x:X \mtimes{} Y;a\#x:X \mwedge{} a\#y:Y)
Date html generated:
2016_05_13-PM-03_21_37
Last ObjectModification:
2015_12_26-AM-09_11_53
Theory : atom_1
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