Nuprl Lemma : symmetric_rel_or
∀[T:Type]. ∀[R1,R2:T ⟶ T ⟶ ℙ]. (Sym(T;x,y.x R1 y) ⇒ Sym(T;x,y.x R2 y) ⇒ Sym(T;x,y.x (R1 ∨ R2) y))
Proof
Definitions occuring in Statement :
rel_or: R1 ∨ R2,
sym: Sym(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rel_or: R1 ∨ R2,
sym: Sym(T;x,y.E[x; y]),
infix_ap: x f y,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
or_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
thin,
inlFormation,
applyEquality,
hypothesisEquality,
inrFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
lambdaEquality,
functionEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R1,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Sym(T;x,y.x R1 y) {}\mRightarrow{} Sym(T;x,y.x R2 y) {}\mRightarrow{} Sym(T;x,y.x (R1 \mvee{} R2) y))
Date html generated:
2019_06_20-PM-00_31_09
Last ObjectModification:
2018_09_26-PM-00_44_10
Theory : relations
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