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Theorem class2set 5352
Description: The class of elements of 𝐴 "such that 𝐴 is a set" is a set. That class is equal to 𝐴 when 𝐴 is a set (see class2seteq 3699) and to the empty set when 𝐴 is a proper class. (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 5330 . 2 (𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} ∈ V)
2 simpl 481 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝑥𝐴) → ¬ 𝐴 ∈ V)
32nrexdv 3147 . . . 4 𝐴 ∈ V → ¬ ∃𝑥𝐴 𝐴 ∈ V)
4 rabn0 4384 . . . . 5 ({𝑥𝐴𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥𝐴 𝐴 ∈ V)
54necon1bbii 2988 . . . 4 (¬ ∃𝑥𝐴 𝐴 ∈ V ↔ {𝑥𝐴𝐴 ∈ V} = ∅)
63, 5sylib 217 . . 3 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = ∅)
7 0ex 5306 . . 3 ∅ ∈ V
86, 7eqeltrdi 2839 . 2 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} ∈ V)
91, 8pm2.61i 182 1 {𝑥𝐴𝐴 ∈ V} ∈ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1539  wcel 2104  wrex 3068  {crab 3430  Vcvv 3472  c0 4321
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322
This theorem is referenced by: (None)
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