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| Mirrors > Home > MPE Home > Th. List > eldifad | Structured version Visualization version GIF version | ||
| Description: If a class is in the difference of two classes, it is also in the minuend. One-way deduction form of eldif 3961. (Contributed by David Moews, 1-May-2017.) |
| Ref | Expression |
|---|---|
| eldifad.1 | ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) |
| Ref | Expression |
|---|---|
| eldifad | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldifad.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐵 ∖ 𝐶)) | |
| 2 | eldif 3961 | . . 3 ⊢ (𝐴 ∈ (𝐵 ∖ 𝐶) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) | |
| 3 | 1, 2 | sylib 218 | . 2 ⊢ (𝜑 → (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶)) |
| 4 | 3 | simpld 494 | 1 ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
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