Nuprl Lemma : binrel_le_antisymmetry
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((R ≡>{T} R') ⇒ (R' ≡>{T} R) ⇒ (R <≡>{T} R'))
Proof
Definitions occuring in Statement :
binrel_le: E ≡>{T} E',
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
binrel_eqv: E <≡>{T} E',
binrel_le: E ≡>{T} E',
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
implies_antisymmetry,
subtype_rel_self,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyEquality,
hypothesisEquality,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
Error :universeIsType,
instantiate,
universeEquality,
because_Cache,
Error :inhabitedIsType,
Error :functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((R \mequiv{}>\{T\} R') {}\mRightarrow{} (R' \mequiv{}>\{T\} R) {}\mRightarrow{} (R <\mequiv{}>\{T\} R'))
Date html generated:
2019_06_20-PM-02_02_24
Last ObjectModification:
2019_01_10-PM-09_36_19
Theory : relations2
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