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| Mirrors > Home > MPE Home > Th. List > eldifsn | Structured version Visualization version GIF version | ||
| Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.) |
| Ref | Expression |
|---|---|
| eldifsn | ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eldif 3961 | . 2 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶})) | |
| 2 | elsng 4640 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → (𝐴 ∈ {𝐶} ↔ 𝐴 = 𝐶)) | |
| 3 | 2 | necon3bbid 2978 | . . 3 ⊢ (𝐴 ∈ 𝐵 → (¬ 𝐴 ∈ {𝐶} ↔ 𝐴 ≠ 𝐶)) |
| 4 | 3 | pm5.32i 574 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
| 5 | 1, 4 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐵 ∖ {𝐶}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ 𝐶)) |
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