Theorem class2set 1747

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.

Assertion

Ref	Expression
class2set	⊢ {x ∈ A∣A ∈ V} ∈ V

Distinct variable group(s):   x,A

Proof of Theorem class2set

Step	Hyp	Ref	Expression
1		rabexg 1705	. 2 ⊢ (A ∈ V → {x ∈ A∣A ∈ V} ∈ V)
2		0ex 1745	. . 3 ⊢ ∅ ∈ V
3		pm3.26 256	. . . . . 6 ⊢ ((¬ A ∈ V ∧ x ∈ A) → ¬ A ∈ V)
4	3	nrexdv 1271	. . . . 5 ⊢ (¬ A ∈ V → ¬ ∃x ∈ A A ∈ V)
5		rabn0 1716	. . . . . 6 ⊢ (¬ {x ∈ A∣A ∈ V} = ∅ ↔ ∃x ∈ A A ∈ V)
6	5	bicon1i 193	. . . . 5 ⊢ (¬ ∃x ∈ A A ∈ V ↔ {x ∈ A∣A ∈ V} = ∅)
7	4, 6	sylib 173	. . . 4 ⊢ (¬ A ∈ V → {x ∈ A∣A ∈ V} = ∅)
8	7	eleq1d 1155	. . 3 ⊢ (¬ A ∈ V → ({x ∈ A∣A ∈ V} ∈ V ↔ ∅ ∈ V))
9	2, 8	mpbiri 169	. 2 ⊢ (¬ A ∈ V → {x ∈ A∣A ∈ V} ∈ V)
10	1, 9	pm2.61i 110	1 ⊢ {x ∈ A∣A ∈ V} ∈ V

Syntax hints:  ¬ wn 1   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  {crab 1204  Vcvv 1348  ∅c0 1707

This theorem is referenced by:  abrexex 2912

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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