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Mirrors > Home > MPE Home > Th. List > class2seteq | Structured version Visualization version GIF version |
Description: Equality theorem based on class2set 5066. (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 3413 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | ax-1 6 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V)) | |
3 | 2 | ralrimiv 3146 | . . . 4 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabid2 3304 | . . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 226 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V}) |
6 | 5 | eqcomd 2783 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
7 | 1, 6 | syl 17 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2106 ∀wral 3089 {crab 3093 Vcvv 3397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-ral 3094 df-rab 3098 df-v 3399 |
This theorem is referenced by: (None) |
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