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Mirrors > Home > MPE Home > Th. List > class2seteq | Structured version Visualization version GIF version |
Description: Writing a set as a class abstraction. This theorem looks artificial but was added to characterize the class abstraction whose existence is proved in class2set 5349. (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 3482 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | ax-1 6 | . . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V)) | |
3 | 2 | ralrimiv 3135 | . 2 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabid2im 3452 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V}) | |
5 | 4 | eqcomd 2731 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
6 | 1, 3, 5 | 3syl 18 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3051 {crab 3419 Vcvv 3463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ral 3052 df-rab 3420 df-v 3465 |
This theorem is referenced by: (None) |
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