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Mirrors > Home > MPE Home > Th. List > elissetv | Structured version Visualization version GIF version |
Description: An element of a class exists. Version of elisset 2821 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2713. Prefer its use over elisset 2821 when sufficient (for instance in usages where 𝑥 is a dummy variable). (Contributed by BJ, 14-Sep-2019.) |
Ref | Expression |
---|---|
elissetv | ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfclel 2815 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
2 | exsimpl 1866 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylbi 217 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = 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 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-clel 2814 |
This theorem is referenced by: elisset 2821 clelab 2885 isseti 3496 elex 3499 elex22 3504 spcimgft 3546 bj-issetiv 36860 bj-ceqsaltv 36870 bj-ceqsalgv 36874 bj-spcimdvv 36879 bj-vtoclg1fv 36902 bj-vtoclg 36903 bj-ru 36927 bj-unexg 37021 |
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