Nuprl Lemma : rset-member_functionality
∀A,B:Set(ℝ). ∀x,y:ℝ.
((∀y:ℝ. SqStable(y ∈ A)) ⇒ (∀y:ℝ. SqStable(y ∈ B)) ⇒ rseteq(A;B) ⇒ {(x ∈ A) ⇒ (y ∈ B)} supposing x = y)
Proof
Definitions occuring in Statement :
rseteq: rseteq(A;B),
rset-member: x ∈ A,
rset: Set(ℝ),
req: x = y,
real: ℝ,
sq_stable: SqStable(P),
uimplies: b supposing a,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
rset-member: x ∈ A,
guard: {T},
rseteq: rseteq(A;B),
rset: Set(ℝ),
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
sq_stable: SqStable(P),
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q
Lemmas referenced :
set_wf,
sq_stable_wf,
iff_wf,
real_wf,
all_wf,
req_wf,
req_witness
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
cut,
introduction,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
rename,
setElimination,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
lambdaEquality,
instantiate,
functionEquality,
cumulativity,
universeEquality,
productElimination
Latex:
\mforall{}A,B:Set(\mBbbR{}). \mforall{}x,y:\mBbbR{}.
((\mforall{}y:\mBbbR{}. SqStable(y \mmember{} A))
{}\mRightarrow{} (\mforall{}y:\mBbbR{}. SqStable(y \mmember{} B))
{}\mRightarrow{} rseteq(A;B)
{}\mRightarrow{} \{(x \mmember{} A) {}\mRightarrow{} (y \mmember{} B)\} supposing x = y)
Date html generated:
2016_05_18-AM-08_08_04
Last ObjectModification:
2016_01_17-AM-02_21_19
Theory : reals
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