Theorem snidb 3760

Description: A class is a set iff it is a member of its singleton. (Contributed by NM, 5-Apr-2004.)

Assertion

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

Proof of Theorem snidb

Step	Hyp	Ref	Expression
1		snidg 3759	. 2 ⊢ (A ∈ V → A ∈ {A})
2		elex 2868	. 2 ⊢ (A ∈ {A} → A ∈ V)
3	1, 2	impbii 180	1 ⊢ (A ∈ V ↔ A ∈ {A})

Syntax hints:   ↔ wb 176   ∈ wcel 1710  Vcvv 2860  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sn 3742

This theorem is referenced by:  snid  3761