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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 2816 with a disjoint variable condition on 𝑉, 𝑥, avoiding df-clab 2711. Prefer its use over elisset 2816 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 2812 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
2 | exsimpl 1872 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉) → ∃𝑥 𝑥 = 𝐴) | |
3 | 1, 2 | sylbi 216 | 1 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-clel 2811 |
This theorem is referenced by: elisset 2816 clelab 2880 isseti 3490 elex22 3497 vtoclgft 3541 bj-elissetALT 35756 bj-issetiv 35757 bj-ceqsaltv 35767 bj-ceqsalgv 35771 bj-spcimdvv 35776 bj-vtoclg1fv 35799 bj-vtoclg 35800 bj-ru 35825 bj-unexg 35919 |
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