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Metamath Proof Explorer

Theorem uspgr1ewop

Description: A simple pseudograph with (at least) two vertices and one edge represented by a singleton word. (Contributed by AV, 9-Jan-2021)

Ref Expression
Assertion uspgr1ewop 
|- ( ( V e. W /\ A e. V /\ B e. V ) -> <. V , <" { A , B } "> >. e. USPGraph )

Proof

Step Hyp Ref Expression
1 prex 
 |-  { A , B } e. _V
2 s1val 
 |-  ( { A , B } e. _V -> <" { A , B } "> = { <. 0 , { A , B } >. } )
3 1 2 mp1i 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> <" { A , B } "> = { <. 0 , { A , B } >. } )
4 3 opeq2d 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> <. V , <" { A , B } "> >. = <. V , { <. 0 , { A , B } >. } >. )
5 simp1 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> V e. W )
6 c0ex 
 |-  0 e. _V
7 6 a1i 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> 0 e. _V )
8 3simpc 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> ( A e. V /\ B e. V ) )
9 uspgr1eop 
 |-  ( ( ( V e. W /\ 0 e. _V ) /\ ( A e. V /\ B e. V ) ) -> <. V , { <. 0 , { A , B } >. } >. e. USPGraph )
10 5 7 8 9 syl21anc 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> <. V , { <. 0 , { A , B } >. } >. e. USPGraph )
11 4 10 eqeltrd 
 |-  ( ( V e. W /\ A e. V /\ B e. V ) -> <. V , <" { A , B } "> >. e. USPGraph )