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	⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V

Distinct variable group:   𝑥,𝐴

Proof of Theorem class2set

Step	Hyp	Ref	Expression
1		rabexg 5330	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V)
2		simpl 483	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V)
3	2	nrexdv 3149	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabn0 4384	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
5	4	necon1bbii 2990	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
6	3, 5	sylib 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
7		0ex 5306	. . 3 ⊢ ∅ ∈ V
8	6, 7	eqeltrdi 2841	. 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V)
9	1, 8	pm2.61i 182	1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  Vcvv 3474  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322

This theorem is referenced by: (None)