Step |
Hyp |
Ref |
Expression |
1 |
|
metf |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ ) |
2 |
|
frel |
⊢ ( 𝐷 : ( 𝑋 × 𝑋 ) ⟶ ℝ → Rel 𝐷 ) |
3 |
|
reldm0 |
⊢ ( Rel 𝐷 → ( 𝐷 = ∅ ↔ dom 𝐷 = ∅ ) ) |
4 |
1 2 3
|
3syl |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( 𝐷 = ∅ ↔ dom 𝐷 = ∅ ) ) |
5 |
1
|
fdmd |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → dom 𝐷 = ( 𝑋 × 𝑋 ) ) |
6 |
5
|
eqeq1d |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( dom 𝐷 = ∅ ↔ ( 𝑋 × 𝑋 ) = ∅ ) ) |
7 |
4 6
|
bitrd |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( 𝐷 = ∅ ↔ ( 𝑋 × 𝑋 ) = ∅ ) ) |
8 |
|
xpeq0 |
⊢ ( ( 𝑋 × 𝑋 ) = ∅ ↔ ( 𝑋 = ∅ ∨ 𝑋 = ∅ ) ) |
9 |
|
oridm |
⊢ ( ( 𝑋 = ∅ ∨ 𝑋 = ∅ ) ↔ 𝑋 = ∅ ) |
10 |
8 9
|
bitri |
⊢ ( ( 𝑋 × 𝑋 ) = ∅ ↔ 𝑋 = ∅ ) |
11 |
7 10
|
bitrdi |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( 𝐷 = ∅ ↔ 𝑋 = ∅ ) ) |
12 |
11
|
necon3bid |
⊢ ( 𝐷 ∈ ( Met ‘ 𝑋 ) → ( 𝐷 ≠ ∅ ↔ 𝑋 ≠ ∅ ) ) |