Theorem abex 5289

Description: Conditions for a class abstraction to be a set. Remark: This proof is shorter than a proof using abexd 5288. (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 4034	. . 3 ⊢ ({𝑥 ∣ 𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑 → 𝑥 ∈ 𝐴))
3		abex.1	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝐴)
4	2, 3	mpgbir 1799	. 2 ⊢ {𝑥 ∣ 𝜑} ⊆ 𝐴
5	1, 4	ssexi 5285	1 ⊢ {𝑥 ∣ 𝜑} ∈ V

Syntax hints:   → wi 4   ∈ wcel 2109  {cab 2708  Vcvv 3455   ⊆ wss 3922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-rab 3412  df-v 3457  df-in 3929  df-ss 3939

This theorem is referenced by:  grimfn  47834  isgrim  47837