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| Mirrors > Home > MPE Home > Th. List > issetlem | Structured version Visualization version GIF version | ||
| Description: Lemma for elisset 2823 and isset 3494. (Contributed by NM, 26-May-1993.) Extract from the proof of isset 3494. (Revised by WL, 2-Feb-2025.) |
| Ref | Expression |
|---|---|
| issetlem.1 | ⊢ 𝑥 ∈ 𝑉 |
| Ref | Expression |
|---|---|
| issetlem | ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfclel 2817 | . 2 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) | |
| 2 | issetlem.1 | . . . 4 ⊢ 𝑥 ∈ 𝑉 | |
| 3 | 2 | biantru 529 | . . 3 ⊢ (𝑥 = 𝐴 ↔ (𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) |
| 4 | 3 | exbii 1848 | . 2 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝑥 ∈ 𝑉)) |
| 5 | 1, 4 | bitr4i 278 | 1 ⊢ (𝐴 ∈ 𝑉 ↔ ∃𝑥 𝑥 = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ 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: isset 3494 |
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