Theorem snidb 1829

Description: A class is a set iff it is a member of its singleton.

Assertion

Ref	Expression
snidb	⊢ (A ∈ V ↔ A ∈ {A})

Proof of Theorem snidb

Step	Hyp	Ref	Expression
1		snidg 1828	. 2 ⊢ (A ∈ V → A ∈ {A})
2		elisset 1354	. 2 ⊢ (A ∈ {A} → A ∈ V)
3	1, 2	impbi 139	1 ⊢ (A ∈ V ↔ A ∈ {A})

Syntax hints:   ↔ wb 127   ∈ wcel 1092  Vcvv 1348  {csn 1808

This theorem is referenced by:  snid 1830

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-16 922  ax-17 925  ax-ext 1074

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-v 1349  df-un 1490  df-sn 1811  df-pr 1812

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