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Theorem abssexg 3991
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 3990 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 df-pw 3417 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
32eleq1i 2150 . . 3 (𝒫 𝐴 ∈ V ↔ {𝑥𝑥𝐴} ∈ V)
4 simpl 107 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
54ss2abi 3082 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
6 ssexg 3953 . . . 4 (({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴} ∧ {𝑥𝑥𝐴} ∈ V) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
75, 6mpan 415 . . 3 ({𝑥𝑥𝐴} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
83, 7sylbi 119 . 2 (𝒫 𝐴 ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
91, 8syl 14 1 (𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wcel 1436  {cab 2071  Vcvv 2615  wss 2988  𝒫 cpw 3415
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001  df-pw 3417
This theorem is referenced by:  pmex  6362
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