Nuprl Lemma : setmem_functionality_1
∀s,x1,x2:coSet{i:l}. (seteq(x1;x2) ⇒ ((x1 ∈ s) ⇐⇒ (x2 ∈ s)))
Proof
Definitions occuring in Statement :
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q
Definitions unfolded in proof :
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T
Lemmas referenced :
seteqweaken_wf,
equal_wf,
coSet_wf,
seteq_wf,
setmemfunc_wf
Rules used in proof :
dependent_functionElimination,
axiomEquality,
instantiate,
functionExtensionality,
sqequalRule,
hypothesis,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
applyEquality,
cut,
lambdaEquality,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
introduction
Latex:
\mforall{}s,x1,x2:coSet\{i:l\}. (seteq(x1;x2) {}\mRightarrow{} ((x1 \mmember{} s) \mLeftarrow{}{}\mRightarrow{} (x2 \mmember{} s)))
Date html generated:
2018_07_29-AM-09_51_41
Last ObjectModification:
2018_07_11-PM-00_37_14
Theory : constructive!set!theory
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