Theorem class2seteq 4142

Description: Equality theorem for classes and sets . (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 2737	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		ax-1 6	. . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
3	2	ralrimiv 2538	. . . 4 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabid2 2642	. . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
5	3, 4	sylibr 133	. . 3 ⊢ (𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V})
6	5	eqcomd 2171	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)
7	1, 6	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = 𝐴)

Syntax hints:   → wi 4   = wceq 1343   ∈ wcel 2136  ∀wral 2444  {crab 2448  Vcvv 2726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-ral 2449  df-rab 2453  df-v 2728

This theorem is referenced by: (None)