This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: The size of the intersection of a set and a class is less than or equal to
the size of the set. (Contributed by AV, 4-Jan-2021)
|
|
Ref |
Expression |
|
Assertion |
hashin |
⊢ ( 𝐴 ∈ 𝑉 → ( ♯ ‘ ( 𝐴 ∩ 𝐵 ) ) ≤ ( ♯ ‘ 𝐴 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
inss1 |
⊢ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 |
| 2 |
|
hashss |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ ( 𝐴 ∩ 𝐵 ) ⊆ 𝐴 ) → ( ♯ ‘ ( 𝐴 ∩ 𝐵 ) ) ≤ ( ♯ ‘ 𝐴 ) ) |
| 3 |
1 2
|
mpan2 |
⊢ ( 𝐴 ∈ 𝑉 → ( ♯ ‘ ( 𝐴 ∩ 𝐵 ) ) ≤ ( ♯ ‘ 𝐴 ) ) |