Theorem abex 5282

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5281. (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 4021	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
4	1, 3	ssexi 5278	1 ⊢ {𝑥 ∣ 𝜑} ∈ V

Syntax hints:   → wi 4   ∈ wcel 2142  {cab 2740  Vcvv 3454

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3456  df-in 3911  df-ss 3921

This theorem is referenced by:  opex  5431  grimfn  48501  isgrim  48504