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Theorem abssexg 3925
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 (A 𝑉 → {x ∣ (xA φ)} V)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   𝑉(x)

Proof of Theorem abssexg
StepHypRef Expression
1 pwexg 3924 . 2 (A 𝑉 → 𝒫 A V)
2 df-pw 3353 . . . 4 𝒫 A = {xxA}
32eleq1i 2100 . . 3 (𝒫 A V ↔ {xxA} V)
4 simpl 102 . . . . 5 ((xA φ) → xA)
54ss2abi 3006 . . . 4 {x ∣ (xA φ)} ⊆ {xxA}
6 ssexg 3887 . . . 4 (({x ∣ (xA φ)} ⊆ {xxA} {xxA} V) → {x ∣ (xA φ)} V)
75, 6mpan 400 . . 3 ({xxA} V → {x ∣ (xA φ)} V)
83, 7sylbi 114 . 2 (𝒫 A V → {x ∣ (xA φ)} V)
91, 8syl 14 1 (A 𝑉 → {x ∣ (xA φ)} V)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  {cab 2023  Vcvv 2551  wss 2911  𝒫 cpw 3351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353
This theorem is referenced by: (None)
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