Theorem class2seteq 3916

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 2566	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		ax-1 5	. . . . 5 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 → 𝐴 ∈ V))
3	2	ralrimiv 2391	. . . 4 ⊢ (𝐴 ∈ V → ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabid2 2486	. . . 4 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ↔ ∀𝑥 ∈ 𝐴 𝐴 ∈ V)
5	3, 4	sylibr 137	. . 3 ⊢ (𝐴 ∈ V → 𝐴 = {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V})
6	5	eqcomd 2045	. 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 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ral 2311  df-rab 2315  df-v 2559

This theorem is referenced by: (None)