Nuprl Lemma : anti_sym_functionality_wrt_iff
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ].
uiff(AntiSym(T;x,y.R[x;y]);AntiSym(T;x,y.R'[x;y])) supposing ∀x,y:T. (R[x;y] ⇐⇒ R'[x;y])
Proof
Definitions occuring in Statement :
anti_sym: AntiSym(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],
all: ∀x:A. B[x],
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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
all_wf,
equal_wf,
uiff_wf,
iff_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
lambdaFormation,
hypothesis,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
axiomEquality,
universeEquality,
because_Cache,
extract_by_obid,
isectElimination,
functionEquality,
addLevel,
productElimination,
independent_isectElimination,
allFunctionality,
independent_functionElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
uiff(AntiSym(T;x,y.R[x;y]);AntiSym(T;x,y.R'[x;y])) supposing \mforall{}x,y:T. (R[x;y] \mLeftarrow{}{}\mRightarrow{} R'[x;y])
Date html generated:
2017_04_14-AM-07_37_50
Last ObjectModification:
2017_02_27-PM-03_10_01
Theory : rel_1
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