Nuprl Lemma : rel_plus_irreflexive
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (WellFnd{i}(T;x,y.x R y) ⇒ (∀x:T. (¬(x R+ x))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
not: ¬A,
false: False,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
infix_ap: x f y,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
not_wf,
rel_plus_wf,
all_wf,
wellfounded_wf,
rel_plus_implies,
wellfounded-irreflexive,
rel_plus_trans,
rel-rel-plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
sqequalHypSubstitution,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
lemma_by_obid,
applyEquality,
hypothesisEquality,
because_Cache,
independent_functionElimination,
voidElimination,
functionEquality,
dependent_functionElimination,
cumulativity,
universeEquality,
isect_memberEquality,
unionElimination,
productElimination,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (WellFnd\{i\}(T;x,y.x R y) {}\mRightarrow{} (\mforall{}x:T. (\mneg{}(x R\msupplus{} x))))
Date html generated:
2016_05_14-PM-03_53_55
Last ObjectModification:
2015_12_26-PM-06_56_27
Theory : relations2
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