Nuprl Lemma : xxconnex_iff_trichot_a
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (connex(T;Ro) ⇐⇒ ∀a,b:T. ((R a b) ∨ (a = b ∈ T) ∨ (R b a)))
Proof
Definitions occuring in Statement :
refl_cl: Eo,
xxconnex: connex(T;R),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
refl_cl: Eo,
xxconnex: connex(T;R),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
connex: Connex(T;x,y.R[x; y]),
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
guard: {T}
Lemmas referenced :
connex_wf,
or_wf,
equal_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
hypothesisEquality,
cut,
lemma_by_obid,
isectElimination,
thin,
lambdaEquality,
hypothesis,
applyEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
unionElimination,
inrFormation,
inlFormation,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (connex(T;R\msupzero{}) \mLeftarrow{}{}\mRightarrow{} \mforall{}a,b:T. ((R a b) \mvee{} (a = b) \mvee{} (R b a)))
Date html generated:
2016_05_15-PM-00_01_54
Last ObjectModification:
2015_12_26-PM-11_25_52
Theory : gen_algebra_1
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