Theorem snidg 3759

Description: A set is a member of its singleton. Part of Theorem 7.6 of [Quine] p. 49. (Contributed by NM, 28-Oct-2003.)

Assertion

Ref	Expression
snidg	⊢ (A ∈ V → A ∈ {A})

Proof of Theorem snidg

Step	Hyp	Ref	Expression
1		eqid 2353	. 2 ⊢ A = A
2		elsncg 3756	. 2 ⊢ (A ∈ V → (A ∈ {A} ↔ A = A))
3	1, 2	mpbiri 224	1 ⊢ (A ∈ V → A ∈ {A})

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {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:  snidb  3760  elsnc2g  3762  snnzg  3834  opkth1g  4131  fvunsn  5445  nchoicelem6  6295  dmfrec  6317  frec0  6322