Theorem elsnc2 3763

Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that B, rather than A, be a set. (Contributed by NM, 12-Jun-1994.)

Hypothesis

Ref	Expression
elsnc2.1	⊢ B ∈ V

Assertion

Ref	Expression
elsnc2	⊢ (A ∈ {B} ↔ A = B)

Proof of Theorem elsnc2

Step	Hyp	Ref	Expression
1		elsnc2.1	. 2 ⊢ B ∈ V
2		elsnc2g 3762	. 2 ⊢ (B ∈ V → (A ∈ {B} ↔ A = B))
3	1, 2	ax-mp 5	1 ⊢ (A ∈ {B} ↔ A = B)

Syntax hints:   ↔ wb 176   = wceq 1642   ∈ 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:  eluni1g  4173  el0c  4422