Theorem class2set 5291

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 3661) 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 5273	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V)
2		simpl 482	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V)
3	2	nrexdv 3125	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabn0 4337	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
5	4	necon1bbii 2975	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
6	3, 5	sylib 218	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
7		0ex 5243	. . 3 ⊢ ∅ ∈ V
8	6, 7	eqeltrdi 2837	. 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 2110  ∃wrex 3054  {crab 3393  Vcvv 3434  ∅c0 4281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-dif 3903  df-in 3907  df-ss 3917  df-nul 4282  df-pw 4550

This theorem is referenced by: (None)