[ BeginEnv "definition"
, Label "universal_set"
, Word "a"
, Word "set"
, BeginEnv "math"
, Variable "V"
, EndEnv "math"
, Word "is"
, Word "universal"
, Word "iff"
, Word "for"
, Word "all"
, Word "sets"
, BeginEnv "math"
, Variable "x"
, EndEnv "math"
, Word "we"
, Word "have"
, BeginEnv "math"
, Variable "x"
, Command "in"
, Variable "V"
, EndEnv "math"
, Symbol "."
, EndEnv "definition"
, BeginEnv "theorem"
, Label "no_universal_set"
, Word "there"
, Word "exists"
, Word "no"
, Word "universal"
, Word "set"
, Symbol "."
, EndEnv "theorem"
, BeginEnv "proof"
, Word "suppose"
, Word "not"
, Symbol "."
, Word "take"
, Word "a"
, Word "universal"
, Word "set"
, BeginEnv "math"
, Variable "V"
, EndEnv "math"
, Symbol "."
, Word "let"
, BeginEnv "math"
, Variable "R"
, Symbol "="
, VisibleBraceL
, Variable "x"
, Command "in"
, Variable "V"
, Command "mid"
, Variable "x"
, Command "not"
, Command "in"
, Variable "x"
, VisibleBraceR
, EndEnv "math"
, Symbol "."
, Word "then"
, BeginEnv "math"
, Variable "R"
, Command "in"
, Variable "R"
, EndEnv "math"
, Word "iff"
, BeginEnv "math"
, Variable "R"
, Command "not"
, Command "in"
, Variable "R"
, EndEnv "math"
, Symbol "."
, Word "contradiction"
, Symbol "."
, EndEnv "proof"
]