Theorem sepab 5302

Description: Separation Scheme (Aussonderung) in terms of a class abstraction. Prefer using the more natural statement rabexg 5307. (Contributed by NM, 8-Jun-1994.) Put in closed form. (Revised by BJ, 18-Jul-2026.)

Assertion

Ref	Expression
sepab	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem sepab

Step	Hyp	Ref	Expression
1		id 23	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
2		ssab2 4032	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴)
4	1, 3	ssexd 5294	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∈ wcel 2142  {cab 2740  Vcvv 3454   ⊆ wss 3904

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-in 3911  df-ss 3921

This theorem is used by:  rabfmpunirn  33009