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Theorem class2set 5376
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 3720) 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 5358 . 2 (𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} ∈ V)
2 simpl 482 . . . . 5 ((¬ 𝐴 ∈ V ∧ 𝑥𝐴) → ¬ 𝐴 ∈ V)
32nrexdv 3151 . . . 4 𝐴 ∈ V → ¬ ∃𝑥𝐴 𝐴 ∈ V)
4 rabn0 4408 . . . . 5 ({𝑥𝐴𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥𝐴 𝐴 ∈ V)
54necon1bbii 2992 . . . 4 (¬ ∃𝑥𝐴 𝐴 ∈ V ↔ {𝑥𝐴𝐴 ∈ V} = ∅)
63, 5sylib 218 . . 3 𝐴 ∈ V → {𝑥𝐴𝐴 ∈ V} = ∅)
7 0ex 5328 . . 3 ∅ ∈ V
86, 7eqeltrdi 2846 . 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 1537  wcel 2103  wrex 3072  {crab 3438  Vcvv 3482  c0 4347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-in 3977  df-ss 3987  df-nul 4348  df-pw 4624
This theorem is referenced by: (None)
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