Theorem class2set 2730

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:   x,A

Proof of Theorem class2set

Step	Hyp	Ref	Expression
1		rabexg 2720	. 2 ⊢ (A ∈ V → {x ∈ A∣A ∈ V} ∈ V)
2		pm3.26 319	. . . . 5 ⊢ ((¬ A ∈ V ⋀ x ∈ A) → ¬ A ∈ V)
3	2	nrexdv 1728	. . . 4 ⊢ (¬ A ∈ V → ¬ ∃x ∈ A A ∈ V)
4		rabn0 2289	. . . . 5 ⊢ ({x ∈ A∣A ∈ V} ≠ ∅ ↔ ∃x ∈ A A ∈ V)
5	4	necon1bbii 1615	. . . 4 ⊢ (¬ ∃x ∈ A A ∈ V ↔ {x ∈ A∣A ∈ V} = ∅)
6	3, 5	sylib 198	. . 3 ⊢ (¬ A ∈ V → {x ∈ A∣A ∈ V} = ∅)
7		0ex 2707	. . 3 ⊢ ∅ ∈ V
8	6, 7	syl6eqel 1554	. 2 ⊢ (¬ A ∈ V → {x ∈ A∣A ∈ V} ∈ V)
9	1, 8	pm2.61i 126	1 ⊢ {x ∈ A∣A ∈ V} ∈ V

Syntax hints:  ¬ wn 2   = wceq 955   ∈ wcel 957  ∃wrex 1644  {crab 1646  Vcvv 1808  ∅c0 2277

This theorem is referenced by:  abrexex 3855  fsum1s 6962  fsump1s 6966

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-nul 2706

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-rab 1650  df-v 1809  df-dif 2046  df-in 2048  df-ss 2050  df-nul 2278

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