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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 3882 | . 2 ⊢ (𝑋 ∈ (𝐴 ∩ 𝐵) ↔ (𝑋 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵)) | |
4 | 1, 2, 3 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐴 ∩ 𝐵)) |
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