Metamath Proof Explorer
Description: The sets in the cartesian product of singletons with other sets, are
disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020)
|
|
Ref |
Expression |
|
Assertion |
disjsnxp |
⊢ Disj 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
sndisj |
⊢ Disj 𝑗 ∈ 𝐴 { 𝑗 } |
2 |
1
|
a1i |
⊢ ( ⊤ → Disj 𝑗 ∈ 𝐴 { 𝑗 } ) |
3 |
2
|
disjxp1 |
⊢ ( ⊤ → Disj 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) ) |
4 |
3
|
mptru |
⊢ Disj 𝑗 ∈ 𝐴 ( { 𝑗 } × 𝐵 ) |