Theorem abssexg 5353

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 5349	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
2		df-pw 4563	. . . 4 ⊢ 𝒫 𝐴 = {𝑥 ∣ 𝑥 ⊆ 𝐴}
3	2	eleq1i 2852	. . 3 ⊢ (𝒫 𝐴 ∈ V ↔ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V)
4		simpl 487	. . . . 5 ⊢ ((𝑥 ⊆ 𝐴 ∧ 𝜑) → 𝑥 ⊆ 𝐴)
5	4	ss2abi 4019	. . . 4 ⊢ {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴}
6		ssexg 5289	. . . 4 ⊢ (({𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ⊆ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∧ {𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V) → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 702	. . 3 ⊢ ({𝑥 ∣ 𝑥 ⊆ 𝐴} ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
8	3, 7	sylbi 220	. 2 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)
9	1, 8	syl 18	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ (𝑥 ⊆ 𝐴 ∧ 𝜑)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2141  {cab 2739  Vcvv 3453   ⊆ wss 3904  𝒫 cpw 4561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pow 5336

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1571  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-in 3911  df-ss 3921  df-pw 4563

This theorem is referenced by:  pmex  8828  tgval  23091