Step |
Hyp |
Ref |
Expression |
1 |
|
usgr1vr |
|- ( ( A e. _V /\ ( Vtx ` G ) = { A } ) -> ( G e. USGraph -> ( iEdg ` G ) = (/) ) ) |
2 |
1
|
adantrl |
|- ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( G e. USGraph -> ( iEdg ` G ) = (/) ) ) |
3 |
|
simplrl |
|- ( ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) /\ ( iEdg ` G ) = (/) ) -> G e. W ) |
4 |
|
simpr |
|- ( ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) /\ ( iEdg ` G ) = (/) ) -> ( iEdg ` G ) = (/) ) |
5 |
3 4
|
usgr0e |
|- ( ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) /\ ( iEdg ` G ) = (/) ) -> G e. USGraph ) |
6 |
5
|
ex |
|- ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( ( iEdg ` G ) = (/) -> G e. USGraph ) ) |
7 |
2 6
|
impbid |
|- ( ( A e. _V /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) |
8 |
7
|
ex |
|- ( A e. _V -> ( ( G e. W /\ ( Vtx ` G ) = { A } ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) ) |
9 |
|
snprc |
|- ( -. A e. _V <-> { A } = (/) ) |
10 |
|
simpl |
|- ( ( G e. W /\ ( Vtx ` G ) = { A } ) -> G e. W ) |
11 |
|
simprr |
|- ( ( { A } = (/) /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( Vtx ` G ) = { A } ) |
12 |
|
simpl |
|- ( ( { A } = (/) /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> { A } = (/) ) |
13 |
11 12
|
eqtrd |
|- ( ( { A } = (/) /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( Vtx ` G ) = (/) ) |
14 |
|
usgr0vb |
|- ( ( G e. W /\ ( Vtx ` G ) = (/) ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) |
15 |
10 13 14
|
syl2an2 |
|- ( ( { A } = (/) /\ ( G e. W /\ ( Vtx ` G ) = { A } ) ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) |
16 |
15
|
ex |
|- ( { A } = (/) -> ( ( G e. W /\ ( Vtx ` G ) = { A } ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) ) |
17 |
9 16
|
sylbi |
|- ( -. A e. _V -> ( ( G e. W /\ ( Vtx ` G ) = { A } ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) ) |
18 |
8 17
|
pm2.61i |
|- ( ( G e. W /\ ( Vtx ` G ) = { A } ) -> ( G e. USGraph <-> ( iEdg ` G ) = (/) ) ) |