Step |
Hyp |
Ref |
Expression |
1 |
|
usgr1vr |
⊢ ( ( 𝐴 ∈ V ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) ) |
2 |
1
|
adantrl |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph → ( iEdg ‘ 𝐺 ) = ∅ ) ) |
3 |
|
simplrl |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ 𝑊 ) |
4 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → ( iEdg ‘ 𝐺 ) = ∅ ) |
5 |
3 4
|
usgr0e |
⊢ ( ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) ∧ ( iEdg ‘ 𝐺 ) = ∅ ) → 𝐺 ∈ USGraph ) |
6 |
5
|
ex |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( ( iEdg ‘ 𝐺 ) = ∅ → 𝐺 ∈ USGraph ) ) |
7 |
2 6
|
impbid |
⊢ ( ( 𝐴 ∈ V ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) |
8 |
7
|
ex |
⊢ ( 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) ) |
9 |
|
snprc |
⊢ ( ¬ 𝐴 ∈ V ↔ { 𝐴 } = ∅ ) |
10 |
|
simpl |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → 𝐺 ∈ 𝑊 ) |
11 |
|
simprr |
⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = { 𝐴 } ) |
12 |
|
simpl |
⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → { 𝐴 } = ∅ ) |
13 |
11 12
|
eqtrd |
⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( Vtx ‘ 𝐺 ) = ∅ ) |
14 |
|
usgr0vb |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = ∅ ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) |
15 |
10 13 14
|
syl2an2 |
⊢ ( ( { 𝐴 } = ∅ ∧ ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) |
16 |
15
|
ex |
⊢ ( { 𝐴 } = ∅ → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) ) |
17 |
9 16
|
sylbi |
⊢ ( ¬ 𝐴 ∈ V → ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) ) |
18 |
8 17
|
pm2.61i |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ ( Vtx ‘ 𝐺 ) = { 𝐴 } ) → ( 𝐺 ∈ USGraph ↔ ( iEdg ‘ 𝐺 ) = ∅ ) ) |