Theorem abex 5262

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5261. (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		abss 4009	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
3		abex.1	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
4	2, 3	mpgbir 1800	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
5	1, 4	ssexi 5258	1 ⊢ {𝑥 ∣ 𝜑} ∈ V

Syntax hints:   → wi 4   ∈ wcel 2111  {cab 2709  Vcvv 3436   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-rab 3396  df-v 3438  df-in 3904  df-ss 3914

This theorem is referenced by:  grimfn  47918  isgrim  47921