MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  class2set Structured version   Visualization version   GIF version

Theorem class2set 5066
Description: Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)
Assertion
Ref Expression
class2set {𝑥𝐴𝐴 ∈ V} ∈ V
Distinct variable group:   𝑥,𝐴

Proof of Theorem class2set
StepHypRef Expression
1 rabexg 5048 . 2 (𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} ∈ V)
2 simpl 476 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝑥𝐴) → ¬ 𝐴 ∈ V)
32nrexdv 3181 . . . 4 𝐴 ∈ V → ¬ ∃𝑥𝐴 𝐴 ∈ V)
4 rabn0 4187 . . . . 5 ({𝑥𝐴𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥𝐴 𝐴 ∈ V)
54necon1bbii 3017 . . . 4 (¬ ∃𝑥𝐴 𝐴 ∈ V ↔ {𝑥𝐴𝐴 ∈ V} = ∅)
63, 5sylib 210 . . 3 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = ∅)
7 0ex 5026 . . 3 ∅ ∈ V
86, 7syl6eqel 2866 . 2 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} ∈ V)
91, 8pm2.61i 177 1 {𝑥𝐴𝐴 ∈ V} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1601  wcel 2106  wrex 3090  {crab 3093  Vcvv 3397  c0 4140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-ext 2753  ax-sep 5017  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-dif 3794  df-in 3798  df-ss 3805  df-nul 4141
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator