### Nuprl Lemma : member_singleton

`∀[T:Type]. ∀a,x:T.  ((x ∈ [a]) `⇐⇒` x = a ∈ T)`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` cons: `[a / b]` nil: `[]` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  l_member: `(x ∈ l)` all: `∀x:A. B[x]` member: `t ∈ T` top: `Top` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` exists: `∃x:A. B[x]` cand: `A c∧ B` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` uimplies: `b supposing a` sq_stable: `SqStable(P)` squash: `↓T` so_apply: `x[s]` rev_implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` le: `A ≤ B` sq_type: `SQType(T)` guard: `{T}` select: `L[n]` cons: `[a / b]` nequal: `a ≠ b ∈ T ` subtype_rel: `A ⊆r B` not: `¬A` less_than': `less_than'(a;b)` true: `True` false: `False` subtract: `n - m` less_than: `a < b`
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isect_memberEquality voidElimination voidEquality hypothesis isect_memberFormation lambdaFormation independent_pairFormation productElimination isectElimination lambdaEquality productEquality setElimination rename because_Cache natural_numberEquality cumulativity hypothesisEquality independent_isectElimination independent_functionElimination imageMemberEquality baseClosed imageElimination universeEquality unionElimination instantiate intEquality dependent_set_memberEquality equalityTransitivity equalitySymmetry addEquality applyEquality minusEquality dependent_pairFormation

Latex:
\mforall{}[T:Type].  \mforall{}a,x:T.    ((x  \mmember{}  [a])  \mLeftarrow{}{}\mRightarrow{}  x  =  a)

Date html generated: 2017_04_14-AM-08_37_24
Last ObjectModification: 2017_02_27-PM-03_29_24

Theory : list_0

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