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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 2108, ax-ext 2695, df-clab 2702. (Revised by Wolf Lammen, 30-Nov-2024.) |
Ref | Expression |
---|---|
elex2 | ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2803 | . 2 ⊢ (𝐴 ∈ 𝐵 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵)) | |
2 | exsimpr 1864 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐵) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-clel 2802 |
This theorem is referenced by: negn0 11642 nocvxmin 27651 itg2addnclem2 37043 risci 37358 dvh1dimat 40815 |
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