Theorem class2seteq 3661

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 5315. (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 3471	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		ax-1 6	. . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
3	2	ralrimiv 3153	. 2 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabid2im 3443	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V})
5	4	eqcomd 2766	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)
6	1, 3, 5	3syl 19	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452

This theorem is used by: (None)