### Nuprl Lemma : set-equal-l_contains

`∀[T:Type]. ∀x,y:T List.  (set-equal(T;x;y) `⇐⇒` x ⊆ y ∧ y ⊆ x)`

Proof

Definitions occuring in Statement :  set-equal: `set-equal(T;x;y)` l_contains: `A ⊆ B` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` l_contains: `A ⊆ B` set-equal: `set-equal(T;x;y)` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` rev_implies: `P `` Q` guard: `{T}`
Lemmas referenced :  l_member_wf all_wf iff_wf l_all_iff l_all_wf list_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut independent_pairFormation introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis because_Cache sqequalRule lambdaEquality productElimination productEquality functionEquality addLevel independent_functionElimination dependent_functionElimination setElimination rename setEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}x,y:T  List.    (set-equal(T;x;y)  \mLeftarrow{}{}\mRightarrow{}  x  \msubseteq{}  y  \mwedge{}  y  \msubseteq{}  x)

Date html generated: 2019_06_20-PM-01_30_21
Last ObjectModification: 2018_08_24-PM-11_35_14

Theory : list_1

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