Nuprl Lemma : setsubset_functionality
∀a,b,a',b':coSet{i:l}. (seteq(a;a') ⇒ seteq(b;b') ⇒ ((a ⊆ b) ⇐⇒ (a' ⊆ b')))
Proof
Definitions occuring in Statement :
setsubset: (a ⊆ b),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_weakening,
setmem_functionality,
seteq_wf,
setmem_wf,
coSet_wf,
all_wf,
iff_wf,
setsubset_wf,
setsubset-iff
Rules used in proof :
impliesLevelFunctionality,
allLevelFunctionality,
allFunctionality,
because_Cache,
functionEquality,
lambdaEquality,
sqequalRule,
instantiate,
isectElimination,
cumulativity,
independent_functionElimination,
hypothesis,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
introduction,
impliesFunctionality,
independent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
addLevel,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b,a',b':coSet\{i:l\}. (seteq(a;a') {}\mRightarrow{} seteq(b;b') {}\mRightarrow{} ((a \msubseteq{} b) \mLeftarrow{}{}\mRightarrow{} (a' \msubseteq{} b')))
Date html generated:
2018_07_29-AM-10_01_21
Last ObjectModification:
2018_07_20-PM-06_23_00
Theory : constructive!set!theory
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