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Mirrors > Home > ILE Home > Th. List > class2seteq | GIF version |
Description: Equality theorem for classes and sets . (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.) |
Ref | Expression |
---|---|
class2seteq | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2763 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | ax-1 6 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V)) | |
3 | 2 | ralrimiv 2562 | . . . 4 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabid2 2667 | . . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 134 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V}) |
6 | 5 | eqcomd 2195 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
7 | 1, 6 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2160 ∀wral 2468 {crab 2472 Vcvv 2752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-11 1517 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-ral 2473 df-rab 2477 df-v 2754 |
This theorem is referenced by: (None) |
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