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Theorem abssexg 5129
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 5126 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 df-pw 4418 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
32eleq1i 2850 . . 3 (𝒫 𝐴 ∈ V ↔ {𝑥𝑥𝐴} ∈ V)
4 simpl 475 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
54ss2abi 3929 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
6 ssexg 5077 . . . 4 (({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴} ∧ {𝑥𝑥𝐴} ∈ V) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
75, 6mpan 677 . . 3 ({𝑥𝑥𝐴} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
83, 7sylbi 209 . 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 387  wcel 2048  {cab 2753  Vcvv 3409  wss 3825  𝒫 cpw 4416
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-sep 5054  ax-pow 5113
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-v 3411  df-in 3832  df-ss 3839  df-pw 4418
This theorem is referenced by:  pmex  8203  tgval  21257
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