Theorem class2set 5354

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 3697) 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 5333	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V)
2		simpl 481	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V)
3	2	nrexdv 3139	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabn0 4386	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
5	4	necon1bbii 2980	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
6	3, 5	sylib 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
7		0ex 5307	. . 3 ⊢ ∅ ∈ V
8	6, 7	eqeltrdi 2833	. 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 1533   ∈ wcel 2098  ∃wrex 3060  {crab 3419  Vcvv 3463  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5299  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3948  df-in 3952  df-ss 3962  df-nul 4324

This theorem is referenced by: (None)