Nuprl Lemma : irrefl_trans_imp_sasym
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (st_anti_sym(T;R)) supposing (trans(T;R) and irrefl(T;R))
Proof
Definitions occuring in Statement :
xxst_anti_sym: st_anti_sym(T;R),
xxirrefl: irrefl(T;R),
xxtrans: trans(T;E),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
xxtrans: trans(T;E),
xxirrefl: irrefl(T;R),
xxst_anti_sym: st_anti_sym(T;R),
trans: Trans(T;x,y.E[x; y]),
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
prop: ℙ
Lemmas referenced :
and_wf,
xxtrans_wf,
xxirrefl_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation,
thin,
productElimination,
lemma_by_obid,
isectElimination,
applyEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
voidElimination,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (st\_anti\_sym(T;R)) supposing (trans(T;R) and irrefl(T;R))
Date html generated:
2016_05_15-PM-00_01_49
Last ObjectModification:
2015_12_26-PM-11_25_56
Theory : gen_algebra_1
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