Theorem abex 5299

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5298. (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 4023	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
4	1, 3	ssexi 5295	1 ⊢ {𝑥 ∣ 𝜑} ∈ V

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  {cab 2743  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-in 3913  df-ss 3923

This theorem is used by:  opex  5447  grimfn  48704  isgrim  48707