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Mirrors > Home > MPE Home > Th. List > class2seteq | Structured version Visualization version GIF version |
Description: Equality theorem based on class2set 5234. (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 3419 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | ax-1 6 | . . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V)) | |
3 | 2 | ralrimiv 3097 | . . . 4 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabid2 3286 | . . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐴 ∈ V) | |
5 | 3, 4 | sylibr 237 | . . 3 ⊢ (𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V}) |
6 | 5 | eqcomd 2740 | . 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 1543 ∈ wcel 2110 ∀wral 3054 {crab 3058 Vcvv 3401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2713 df-cleq 2726 df-clel 2812 df-ral 3059 df-rab 3063 df-v 3403 |
This theorem is referenced by: (None) |
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