Nuprl Lemma : in-open-isect
∀[X:Type]. ∀[x:X]. ∀[A,B:Open(X)]. (x ∈ open-isect(A;B) ⇐⇒ x ∈ A ∧ x ∈ B)
Proof
Definitions occuring in Statement :
in-open: x ∈ A,
open-isect: open-isect(A;B),
Open: Open(X),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
open-isect: open-isect(A;B),
in-open: x ∈ A,
Open: Open(X),
prop: ℙ,
rev_implies: P ⇐ Q,
cand: A c∧ B
Lemmas referenced :
sp-meet-is-top,
in-open_wf,
open-isect_wf,
and_wf,
Open_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
lemma_by_obid,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[x:X]. \mforall{}[A,B:Open(X)]. (x \mmember{} open-isect(A;B) \mLeftarrow{}{}\mRightarrow{} x \mmember{} A \mwedge{} x \mmember{} B)
Date html generated:
2019_10_31-AM-07_18_59
Last ObjectModification:
2015_12_28-AM-11_21_44
Theory : synthetic!topology
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