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| Description: If a class is a set, then it is a member of a set. (Contributed by NM, 4-Jan-2002.) Generalize from the proof of elALT 5444. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5295, ax-nul 5305, ax-pow 5364. (Revised by BTernaryTau, 15-Jan-2025.) | 
| Ref | Expression | 
|---|---|
| sels | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eleq1 2828 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
| 2 | 1 | exbidv 1920 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝐴 ∈ 𝑥)) | 
| 3 | el 5441 | . 2 ⊢ ∃𝑥 𝑦 ∈ 𝑥 | |
| 4 | 2, 3 | vtoclg 3553 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∃wex 1778 ∈ wcel 2107 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 | 
| This theorem is referenced by: sat1el2xp 35385 | 
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