### Nuprl Lemma : b-union-equality-disjoint

`∀A,B:Type. ∀a:A. ∀b:B.  ((¬A ⋂ B) `` (¬(a = b ∈ (A ⋃ B))))`

Proof

Definitions occuring in Statement :  isect2: `T1 ⋂ T2` b-union: `A ⋃ B` all: `∀x:A. B[x]` not: `¬A` implies: `P `` Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` bfalse: `ff` ifthenelse: `if b then t else f fi ` btrue: `tt` it: `⋅` unit: `Unit` bool: `𝔹` isect2: `T1 ⋂ T2` member: `t ∈ T` false: `False` not: `¬A` implies: `P `` Q` all: `∀x:A. B[x]` tunion: `⋃x:A.B[x]` b-union: `A ⋃ B` pi2: `snd(t)` so_lambda: `λ2x.t[x]` so_apply: `x[s]` top: `Top` guard: `{T}` uimplies: `b supposing a` pi1: `fst(t)` or: `P ∨ Q` sq_type: `SQType(T)` uiff: `uiff(P;Q)` and: `P ∧ Q`
Lemmas referenced :  istype-universe istype-void isect2_wf subtype_rel_b-union-right subtype_rel_b-union-left b-union_wf bool_wf bfalse_wf ifthenelse_wf pi2_wf top_wf istype-top pair-eta subtype_rel_product pi1_wf bool_cases subtype_base_sq bool_subtype_base eqtt_to_assert eqff_to_assert assert_of_bnot
Rules used in proof :  universeEquality instantiate Error :inhabitedIsType,  Error :functionIsType,  applyEquality isectElimination Error :universeIsType,  Error :equalityIstype,  because_Cache voidElimination extract_by_obid equalitySymmetry equalityTransitivity hypothesisEquality sqequalRule equalityElimination unionElimination isect_memberEquality introduction independent_functionElimination sqequalHypSubstitution hypothesis thin cut Error :lambdaFormation_alt,  sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution imageEqInduction baseClosed Error :dependent_pairEquality_alt,  Error :lambdaEquality_alt,  Error :productIsType,  independent_pairEquality Error :isect_memberEquality_alt,  independent_isectElimination applyLambdaEquality dependent_functionElimination cumulativity productElimination

Latex:
\mforall{}A,B:Type.  \mforall{}a:A.  \mforall{}b:B.    ((\mneg{}A  \mcap{}  B)  {}\mRightarrow{}  (\mneg{}(a  =  b)))

Date html generated: 2019_06_20-PM-00_28_01
Last ObjectModification: 2019_01_02-PM-03_16_14

Theory : subtype_1

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