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Theorem abssexg 4777
 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 (𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem abssexg
StepHypRef Expression
1 pwexg 4776 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 df-pw 4110 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
32eleq1i 2679 . . 3 (𝒫 𝐴 ∈ V ↔ {𝑥𝑥𝐴} ∈ V)
4 simpl 472 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
54ss2abi 3637 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
6 ssexg 4732 . . . 4 (({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴} ∧ {𝑥𝑥𝐴} ∈ V) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
75, 6mpan 702 . . 3 ({𝑥𝑥𝐴} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
83, 7sylbi 206 . 2 (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
91, 8syl 17 1 (𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {cab 2596  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-pow 4769 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by:  pmex  7749  tgval  20570
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