Theorem abssexg 5357

Description: Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
abssexg	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem abssexg

Step	Hyp	Ref	Expression
1		pwexg 5353	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		df-pw 4582	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
3	2	eleq1i 2826	. . 3 ⊢ (𝒫 𝐴 ∈ V ↔ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V)
4		simpl 482	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝑥 ⊆ 𝐴)
5	4	ss2abi 4047	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴}
6		ssexg 5298	. . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∧ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 690	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
8	3, 7	sylbi 217	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
9	1, 8	syl 17	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  {cab 2714  Vcvv 3464   ⊆ wss 3931  𝒫 cpw 4580

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-ext 2708  ax-sep 5271  ax-pow 5340

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-in 3938  df-ss 3948  df-pw 4582

This theorem is referenced by:  pmex  8850  tgval  22898