MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfausab Structured version   Visualization version   GIF version

Theorem zfausab 5274
Description: Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
Hypothesis
Ref Expression
zfausab.1 𝐴 ∈ V
Assertion
Ref Expression
zfausab {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem zfausab
StepHypRef Expression
1 zfausab.1 . 2 𝐴 ∈ V
2 ssab2 4024 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} ⊆ 𝐴
31, 2ssexi 5266 1 {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 396  wcel 2105  {cab 2713  Vcvv 3441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-sep 5243
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3404  df-v 3443  df-in 3905  df-ss 3915
This theorem is referenced by:  rabfmpunirn  31277
  Copyright terms: Public domain W3C validator