Theorem abex 5297

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5296. (Contributed by AV, 19-Apr-2025.)

Hypotheses

Ref	Expression
abex.1	⊢ (𝜑 → 𝑥 ∈ 𝐴)
abex.2	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
abex	⊢ {𝑥 ∣ 𝜑} ∈ V

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abex

Step	Hyp	Ref	Expression
1		abex.2	. 2 ⊢ 𝐴 ∈ V
2		abex.1	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
3	2	abssi 4022	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
4	1, 3	ssexi 5293	1 ⊢ {𝑥 ∣ 𝜑} ∈ V

Syntax hints:   → wi 4   ∈ wcel 2143  {cab 2741  Vcvv 3455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3912  df-ss 3922

This theorem is referenced by:  opex  5445  grimfn  48644  isgrim  48647