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Theorem abssexg 5273
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 5270 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 df-pw 4537 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
32eleq1i 2900 . . 3 (𝒫 𝐴 ∈ V ↔ {𝑥𝑥𝐴} ∈ V)
4 simpl 483 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
54ss2abi 4040 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
6 ssexg 5218 . . . 4 (({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴} ∧ {𝑥𝑥𝐴} ∈ V) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
75, 6mpan 686 . . 3 ({𝑥𝑥𝐴} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
83, 7sylbi 218 . 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 396  wcel 2105  {cab 2796  Vcvv 3492  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-pow 5257
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  pmex  8400  tgval  21491
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