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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 ) |