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Mirrors > Home > ILE Home > Th. List > eqvisset | GIF version |
Description: A class equal to a variable is a set. Note the absence of dv condition, contrary to isset 2605 and issetri 2608. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
eqvisset | ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2604 | . 2 ⊢ 𝑥 ∈ V | |
2 | eleq1 2141 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V)) | |
3 | 1, 2 | mpbii 146 | 1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∈ wcel 1433 Vcvv 2601 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-v 2603 |
This theorem is referenced by: elxp5 4829 xpsnen 6318 |
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