Theorem abssexg 3925

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 3924	. 2 ⊢ (A ∈ 𝑉 → 𝒫 A ∈ V)
2		df-pw 3353	. . . 4 ⊢ 𝒫 A = {x ∣ x ⊆ A}
3	2	eleq1i 2100	. . 3 ⊢ (𝒫 A ∈ V ↔ {x ∣ x ⊆ A} ∈ V)
4		simpl 102	. . . . 5 ⊢ ((x ⊆ A ∧ φ) → x ⊆ A)
5	4	ss2abi 3006	. . . 4 ⊢ {x ∣ (x ⊆ A ∧ φ)} ⊆ {x ∣ x ⊆ A}
6		ssexg 3887	. . . 4 ⊢ (({x ∣ (x ⊆ A ∧ φ)} ⊆ {x ∣ x ⊆ A} ∧ {x ∣ x ⊆ A} ∈ V) → {x ∣ (x ⊆ A ∧ φ)} ∈ V)
7	5, 6	mpan 400	. . 3 ⊢ ({x ∣ x ⊆ A} ∈ V → {x ∣ (x ⊆ A ∧ φ)} ∈ V)
8	3, 7	sylbi 114	. 2 ⊢ (𝒫 A ∈ V → {x ∣ (x ⊆ A ∧ φ)} ∈ V)
9	1, 8	syl 14	1 ⊢ (A ∈ 𝑉 → {x ∣ (x ⊆ A ∧ φ)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  {cab 2023  Vcvv 2551   ⊆ wss 2911  𝒫 cpw 3351

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353

This theorem is referenced by: (None)