Nuprl Lemma : sp-meet_wf

`∀[f,g:Sierpinski].  (f ∧ g ∈ Sierpinski)`

Proof

Definitions occuring in Statement :  sp-meet: `f ∧ g` Sierpinski: `Sierpinski` uall: `∀[x:A]. B[x]` member: `t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` Sierpinski: `Sierpinski` quotient: `x,y:A//B[x; y]` and: `P ∧ Q` sp-meet: `f ∧ g` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` uimplies: `b supposing a` iff: `P `⇐⇒` Q` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` uiff: `uiff(P;Q)` bfalse: `ff` band: `p ∧b q` ifthenelse: `if b then t else f fi ` rev_uimplies: `rev_uimplies(P;Q)` not: `¬A` false: `False` rev_implies: `P `` Q` btrue: `tt` assert: `↑b` true: `True` subtype_rel: `A ⊆r B`
Lemmas referenced :  Sierpinski_wf quotient-member-eq nat_wf bool_wf iff_wf equal-wf-T-base two-class-equiv-rel Sierpinski-bottom_wf coded-pair_wf bool_cases subtype_base_sq bool_subtype_base eqtt_to_assert band_wf btrue_wf bfalse_wf istype-nat equal-Sierpinski-bottom istype-assert bool_cases_sqequal assert_elim not_assert_elim btrue_neq_bfalse code-pair_wf iff_weakening_uiff assert_wf assert_functionality_wrt_uiff coded-code-pair
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution pointwiseFunctionalityForEquality extract_by_obid hypothesis sqequalRule pertypeElimination promote_hyp thin productElimination isectElimination functionEquality lambdaEquality_alt hypothesisEquality baseClosed because_Cache inhabitedIsType equalityTransitivity equalitySymmetry independent_isectElimination dependent_functionElimination lambdaFormation_alt applyEquality unionElimination instantiate cumulativity independent_functionElimination equalityIstype independent_pairFormation rename voidElimination sqequalBase productIsType functionIsType universeIsType axiomEquality isect_memberEquality_alt isectIsTypeImplies natural_numberEquality dependent_set_memberEquality_alt baseApply closedConclusion applyLambdaEquality setElimination independent_pairEquality spreadEquality

Latex:
\mforall{}[f,g:Sierpinski].    (f  \mwedge{}  g  \mmember{}  Sierpinski)

Date html generated: 2019_10_31-AM-06_35_41
Last ObjectModification: 2018_12_13-PM-03_00_20

Theory : synthetic!topology

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