Nuprl Lemma : seteq-iff
∀s1,s2:Set{i:l}. (seteq(s1;s2) ⇐⇒ ∀x:Set{i:l}. ((x ∈ s1) ⇐⇒ (x ∈ s2)))
Proof
Definitions occuring in Statement :
Set: Set{i:l},
setmem: (x ∈ s),
seteq: seteq(s1;s2),
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
guard: {T},
exists: ∃x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
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 :
coSet_wf,
coSet-mem-Set-implies-Set,
co-seteq-iff,
setmem_wf,
iff_wf,
Set_wf,
all_wf,
set-subtype-coSet,
seteq_wf
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
dependent_pairFormation,
independent_isectElimination,
independent_functionElimination,
productElimination,
dependent_functionElimination,
cumulativity,
lambdaEquality,
instantiate,
because_Cache,
sqequalRule,
hypothesis,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}s1,s2:Set\{i:l\}. (seteq(s1;s2) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:Set\{i:l\}. ((x \mmember{} s1) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} s2)))
Date html generated:
2018_07_29-AM-09_51_48
Last ObjectModification:
2018_07_11-PM-02_41_24
Theory : constructive!set!theory
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