A no-spill drinking vessel includes a cup, lid, spout and valve assembly. The valve assembly includes: a liquid-out valve with hinged flap, air vent check valve, air and liquid inlets, air and liquid outlets, and a mechanism for retention to the lid. The valve assembly is constructed from inexpensive molded elastomeric components. In usage mode these components cleave sealingly together by the resilient returning forces they exert on each other's complementary interlocked geometries. The fluid and air passageways, inlet and outlet port openings, mouthpiece channels and openings, and the interior volumes of the valve bodies are defined by spaces between the abutted elastomeric components. When manually separated into a cleaning configuration, the interior liquid-contacting surfaces become exposed, external surfaces for ready cleaning. The closure is baby-proof and small parts are tethered together.