Nuprl Lemma : setmemfunc_wf
∀[x1,s1,x2,s2:coSet{i:l}]. (setmemfunc(x1; s1; x2; s2) ∈ seteq(x1;x2) ⇒ seteq(s1;s2) ⇒ {(x1 ∈ s1) ⇐⇒ (x2 ∈ s2)})
Proof
Definitions occuring in Statement :
setmemfunc: setmemfunc(x1; s1; x2; s2),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
member: t ∈ T
Definitions unfolded in proof :
and: P ∧ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
so_apply: x[s],
guard: {T},
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
setmemfunclemma_ext,
setmemfunc: setmemfunc(x1; s1; x2; s2),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
setmem_wf,
iff_wf,
seteq_wf,
all_wf,
coSet_wf,
subtype_rel_self,
setmemfunclemma_ext
Rules used in proof :
because_Cache,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesisEquality,
cumulativity,
lambdaEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[x1,s1,x2,s2:coSet\{i:l\}].
(setmemfunc(x1; s1; x2; s2) \mmember{} seteq(x1;x2) {}\mRightarrow{} seteq(s1;s2) {}\mRightarrow{} \{(x1 \mmember{} s1) \mLeftarrow{}{}\mRightarrow{} (x2 \mmember{} s2)\})
Date html generated:
2018_07_29-AM-09_51_36
Last ObjectModification:
2018_07_11-PM-00_35_12
Theory : constructive!set!theory
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