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Theorem abssexg 4076
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 4074 . 2 (𝐴𝑉 → 𝒫 𝐴 ∈ V)
2 df-pw 3482 . . . 4 𝒫 𝐴 = {𝑥𝑥𝐴}
32eleq1i 2183 . . 3 (𝒫 𝐴 ∈ V ↔ {𝑥𝑥𝐴} ∈ V)
4 simpl 108 . . . . 5 ((𝑥𝐴𝜑) → 𝑥𝐴)
54ss2abi 3139 . . . 4 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴}
6 ssexg 4037 . . . 4 (({𝑥 ∣ (𝑥𝐴𝜑)} ⊆ {𝑥𝑥𝐴} ∧ {𝑥𝑥𝐴} ∈ V) → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
75, 6mpan 420 . . 3 ({𝑥𝑥𝐴} ∈ V → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)
83, 7sylbi 120 . 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 103  wcel 1465  {cab 2103  Vcvv 2660  wss 3041  𝒫 cpw 3480
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482
This theorem is referenced by:  pmex  6515  tgval  12145
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