Theorem class2set 5280

Description: Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (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 5259	. 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V)
2		simpl 483	. . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V)
3	2	nrexdv 3200	. . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
4		rabn0 4325	. . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V)
5	4	necon1bbii 2995	. . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
6	3, 5	sylib 217	. . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅)
7		0ex 5235	. . 3 ⊢ ∅ ∈ V
8	6, 7	eqeltrdi 2849	. 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 1542   ∈ wcel 2110  ∃wrex 3067  {crab 3070  Vcvv 3431  ∅c0 4262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-in 3899  df-ss 3909  df-nul 4263

This theorem is referenced by: (None)