| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > elind | Structured version Visualization version GIF version | ||
| Description: Deduce membership in an intersection of two classes. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
| Ref | Expression |
|---|---|
| elind.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
| elind.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| Ref | Expression |
|---|---|
| elind | ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elind.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
| 2 | elind.2 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 3 | elin 3967 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
| 4 | 1, 2, 3 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
| Copyright terms: Public domain | W3C validator |