### Nuprl Lemma : quotient-member-eq

`∀[T:Type]. ∀[E:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (E[x;y] `` (x = y ∈ (x,y:T//E[x;y]))) supposing EquivRel(T;x,y.E[x;y])`

Proof

Definitions occuring in Statement :  equiv_rel: `EquivRel(T;x,y.E[x; y])` quotient: `x,y:A//B[x; y]` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` so_apply: `x[s1;s2]` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` cand: `A c∧ B` squash: `↓T` guard: `{T}` true: `True` equiv_rel: `EquivRel(T;x,y.E[x; y])` refl: `Refl(T;x,y.E[x; y])`
Lemmas referenced :  subtype_rel_self equiv_rel_wf istype-universe quotient_wf equal_wf squash_wf true_wf subtype_quotient equal_functionality_wrt_subtype_rel2 equal-wf-base-T subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut Error :lambdaFormation_alt,  hypothesis Error :universeIsType,  applyEquality hypothesisEquality thin sqequalRule instantiate extract_by_obid sqequalHypSubstitution isectElimination universeEquality Error :inhabitedIsType,  Error :lambdaEquality_alt,  dependent_functionElimination axiomEquality Error :functionIsTypeImplies,  Error :isect_memberEquality_alt,  Error :isectIsTypeImplies,  Error :functionIsType,  because_Cache pointwiseFunctionality pertypeMemberEquality equalityTransitivity equalitySymmetry independent_isectElimination independent_pairFormation imageElimination independent_functionElimination natural_numberEquality imageMemberEquality baseClosed Error :dependent_set_memberEquality_alt,  Error :productIsType,  Error :equalityIsType3,  applyLambdaEquality setElimination rename productElimination Error :equalityIsType1,  hyp_replacement

Latex:
\mforall{}[T:Type].  \mforall{}[E:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}x,y:T.    (E[x;y]  {}\mRightarrow{}  (x  =  y))  supposing  EquivRel(T;x,y.E[x;y])

Date html generated: 2019_06_20-PM-00_32_07
Last ObjectModification: 2018_11_24-AM-09_34_59

Theory : quot_1

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