Theorem class2seteq 3666

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 5325. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)

Assertion

Ref	Expression
class2seteq	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem class2seteq

Step	Hyp	Ref	Expression
1		elex 3474	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		ax-1 6	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
3	2	ralrimiv 3154	. 2 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabid2im 3446	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V})
5	4	eqcomd 2767	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)
6	1, 3, 5	3syl 19	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3415  df-v 3455

This theorem is referenced by: (None)