Nuprl Lemma : psub_antisymmetry
∀[P,Q:formula()]. (P = Q ∈ formula()) supposing (Q ⊆ P and P ⊆ Q)
Proof
Definitions occuring in Statement :
psub: a ⊆ b,
formula: formula(),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
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,
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ
Lemmas referenced :
formula_wf,
psub_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
prank-psub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
because_Cache,
unionElimination,
equalitySymmetry,
imageElimination,
productElimination,
isectElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
axiomEquality,
equalityTransitivity
Latex:
\mforall{}[P,Q:formula()]. (P = Q) supposing (Q \msubseteq{} P and P \msubseteq{} Q)
Date html generated:
2016_05_15-PM-07_14_25
Last ObjectModification:
2016_01_16-AM-09_44_01
Theory : general
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