Nuprl Lemma : two-class-equiv-rel
∀[T:Type]. ∀[t:T]. EquivRel(T;x,y.x = t ∈ T ⇐⇒ y = t ∈ T)
Proof
Definitions occuring in Statement :
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
cand: A c∧ B
Lemmas referenced :
equal_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[t:T]. EquivRel(T;x,y.x = t \mLeftarrow{}{}\mRightarrow{} y = t)
Date html generated:
2019_10_31-AM-06_35_22
Last ObjectModification:
2017_07_28-AM-09_11_55
Theory : synthetic!topology
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