Theorem elsncg 3756

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
elsncg	⊢ (A ∈ V → (A ∈ {B} ↔ A = B))

Proof of Theorem elsncg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2359	. 2 ⊢ (x = A → (x = B ↔ A = B))
2		df-sn 3742	. 2 ⊢ {B} = {x ∣ x = B}
3	1, 2	elab2g 2988	1 ⊢ (A ∈ V → (A ∈ {B} ↔ A = B))

Syntax hints:   → wi 4   ↔ wb 176   = 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:  elsnc  3757  elsni  3758  snidg  3759  eltpg  3770  eldifsn  3840  opkth1g  4131