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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 disjoint variable condition, contrary to isset 2736 and issetri 2739. (Contributed by BJ, 27-Apr-2019.) |
Ref | Expression |
---|---|
eqvisset | ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2733 | . 2 ⊢ 𝑥 ∈ V | |
2 | eleq1 2233 | . 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ V ↔ 𝐴 ∈ V)) | |
3 | 1, 2 | mpbii 147 | 1 ⊢ (𝑥 = 𝐴 → 𝐴 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 Vcvv 2730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-v 2732 |
This theorem is referenced by: elxp5 5099 xpsnen 6799 fival 6947 |
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