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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 2823 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2715. Prefer its use over elisset 2823 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 2817 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
| 2 | exsimpl 1868 | . 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 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: elisset 2823 clelab 2887 isseti 3498 elex 3501 elex22 3506 spcimgft 3546 bj-issetiv 36878 bj-ceqsaltv 36888 bj-ceqsalgv 36892 bj-spcimdvv 36897 bj-vtoclg1fv 36920 bj-vtoclg 36921 bj-ru 36945 bj-unexg 37039 |
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