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Mirrors > Home > MPE Home > Th. List > sels | Structured version Visualization version GIF version |
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 5451. (Revised by BJ, 3-Apr-2019.) Avoid ax-sep 5302, ax-nul 5312, ax-pow 5371. (Revised by BTernaryTau, 15-Jan-2025.) |
Ref | Expression |
---|---|
sels | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2827 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥)) | |
2 | 1 | exbidv 1919 | . 2 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 ∈ 𝑥 ↔ ∃𝑥 𝐴 ∈ 𝑥)) |
3 | el 5448 | . 2 ⊢ ∃𝑥 𝑦 ∈ 𝑥 | |
4 | 2, 3 | vtoclg 3554 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝐴 ∈ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∃wex 1776 ∈ wcel 2106 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 |
This theorem is referenced by: sat1el2xp 35364 |
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