Metamath Proof Explorer
Description: A binary relation does not contain the empty set. (Contributed by AV, 15-Nov-2021) (Revised by BJ, 14-Jul-2023)
|
|
Ref |
Expression |
|
Assertion |
0nelrel0 |
⊢ ( Rel 𝑅 → ¬ ∅ ∈ 𝑅 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
df-rel |
⊢ ( Rel 𝑅 ↔ 𝑅 ⊆ ( V × V ) ) |
2 |
1
|
biimpi |
⊢ ( Rel 𝑅 → 𝑅 ⊆ ( V × V ) ) |
3 |
|
0nelxp |
⊢ ¬ ∅ ∈ ( V × V ) |
4 |
3
|
a1i |
⊢ ( Rel 𝑅 → ¬ ∅ ∈ ( V × V ) ) |
5 |
2 4
|
ssneldd |
⊢ ( Rel 𝑅 → ¬ ∅ ∈ 𝑅 ) |