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| Mirrors > Home > MPE Home > Th. List > elex | Structured version Visualization version GIF version | ||
| Description: If a class is a member of another class, then it is a set. Theorem 6.12 of [Quine] p. 44. (Contributed by NM, 26-May-1993.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by Wolf Lammen, 28-May-2025.) |
| Ref | Expression |
|---|---|
| elex | ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elissetv 2822 | . 2 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴) | |
| 2 | isset 3494 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
| 3 | 1, 2 | sylibr 234 | 1 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) |
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