Nuprl Lemma : uanti_sym_functionality_wrt_iff
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
uiff(UniformlyAntiSym(T;x,y.R[x;y]);UniformlyAntiSym(T;x,y.R'[x;y])) supposing ∀[x,y:T]. (R[x;y] ⇐⇒ R'[x;y])
Proof
Definitions occuring in Statement :
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y]),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
guard: {T},
uanti_sym: UniformlyAntiSym(T;x,y.R[x; y])
Lemmas referenced :
uall_wf,
isect_wf,
equal_wf,
uiff_wf,
iff_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
lambdaEquality,
addLevel,
productElimination,
independent_isectElimination,
uallFunctionality,
independent_functionElimination,
universeEquality,
functionEquality,
independent_pairEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
uiff(UniformlyAntiSym(T;x,y.R[x;y]);UniformlyAntiSym(T;x,y.R'[x;y]))
supposing \mforall{}[x,y:T]. (R[x;y] \mLeftarrow{}{}\mRightarrow{} R'[x;y])
Date html generated:
2016_10_21-AM-09_42_13
Last ObjectModification:
2016_08_01-PM-09_49_13
Theory : rel_1
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