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| Mirrors > Home > MPE Home > Th. List > elex2 | Structured version Visualization version GIF version | ||
| Description: If a class contains another class, then it contains some set. (Contributed by Alan Sare, 25-Sep-2011.) Avoid ax-9 2118, ax-ext 2708, df-clab 2715. (Revised by Wolf Lammen, 30-Nov-2024.) |
| Ref | Expression |
|---|---|
| elex2 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfclel 2817 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
| 2 | exsimpr 1869 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐵) | |
| 3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-clel 2816 |
| This theorem is referenced by: negn0 11692 nocvxmin 27823 itg2addnclem2 37679 risci 37994 dvh1dimat 41443 |
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