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| Mirrors > Home > MPE Home > Th. List > sselid | Structured version Visualization version GIF version | ||
| Description: Membership inference from subclass relationship. (Contributed by NM, 25-Jun-2014.) |
| Ref | Expression |
|---|---|
| sseli.1 | ⊢ 𝐴 ⊆ 𝐵 |
| sselid.2 | ⊢ (𝜑 → 𝐶 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| sselid | ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | sselid.2 | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴) | |
| 2 | sseli.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
| 3 | 2 | sseli 3979 | . 2 ⊢ (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝜑 → 𝐶 ∈ 𝐵) |
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