Theorem upgredgssspr

Description: The set of edges of a pseudograph is a subset of the set of unordered pairs of vertices. (Contributed by AV, 24-Nov-2021)

	Ref	Expression
Assertion	upgredgssspr	|- ( G e. UPGraph -> ( Edg ` G ) C_ ( Pairs ` ( Vtx ` G ) ) )

Proof

Step	Hyp	Ref	Expression
1		upgredgss	|- ( G e. UPGraph -> ( Edg ` G ) C_ { p e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } )
2		fvex	|- ( Vtx ` G ) e. _V
3		sprvalpwle2	|- ( ( Vtx ` G ) e. _V -> ( Pairs ` ( Vtx ` G ) ) = { p e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 } )
4	2 3	ax-mp	|- ( Pairs ` ( Vtx ` G ) ) = { p e. ( ~P ( Vtx ` G ) \ { (/) } ) | ( # ` p ) <_ 2 }
5	1 4	sseqtrrdi	|- ( G e. UPGraph -> ( Edg ` G ) C_ ( Pairs ` ( Vtx ` G ) ) )