Nuprl Lemma : seteq-iff-setsubset
∀a,b:coSet{i:l}. (seteq(a;b) ⇐⇒ (a ⊆ b) ∧ (b ⊆ a))
Proof
Definitions occuring in Statement :
setsubset: (a ⊆ b),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q
Definitions unfolded in proof :
guard: {T},
so_apply: x[s],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
setsubset_wf,
seteq_wf,
setsubset-iff,
co-seteq-iff,
iff_wf,
coSet_wf,
all_wf,
setmem_wf
Rules used in proof :
andLevelFunctionality,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
addLevel,
functionEquality,
productEquality,
productElimination,
cumulativity,
lambdaEquality,
sqequalRule,
instantiate,
because_Cache,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}a,b:coSet\{i:l\}. (seteq(a;b) \mLeftarrow{}{}\mRightarrow{} (a \msubseteq{} b) \mwedge{} (b \msubseteq{} a))
Date html generated:
2018_07_29-AM-10_01_24
Last ObjectModification:
2018_07_18-PM-01_30_00
Theory : constructive!set!theory
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