Nuprl Lemma : f-subset_antisymmetry
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[xs,ys:fset(T)]. (xs = ys ∈ fset(T)) supposing (ys ⊆ xs and xs ⊆ ys)
Proof
Definitions occuring in Statement :
f-subset: xs ⊆ ys,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
f-subset: xs ⊆ ys,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
all: ∀x:A. B[x]
Lemmas referenced :
fset-extensionality,
fset-member_witness,
fset-member_wf,
all_wf,
isect_wf,
fset_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
independent_isectElimination,
independent_pairFormation,
because_Cache,
independent_functionElimination,
hypothesis,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
axiomEquality,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[xs,ys:fset(T)]. (xs = ys) supposing (ys \msubseteq{} xs and xs \msubseteq{} ys)
Date html generated:
2016_05_14-PM-03_38_26
Last ObjectModification:
2015_12_26-PM-06_42_15
Theory : finite!sets
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