Theorem elsn2 4158

Description: There is exactly 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
elsn2.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
elsn2	⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Proof of Theorem elsn2

Step	Hyp	Ref	Expression
1		elsn2.1	. 2 ⊢ 𝐵 ∈ V
2		elsn2g 4157	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126

This theorem is referenced by:  fparlem1  7164  fparlem2  7165  el1o  7466  fin1a2lem11  9115  fin1a2lem12  9116  elnn0  11171  elxnn0  11242  elfzp1  12261  fsumss  14303  fprodss  14517  elhoma  16505  islpidl  19067  zrhrhmb  19678  rest0  20783  qustgphaus  21736  taylfval  23917  elch0  27495  atoml2i  28626  bj-eltag  32158  bj-rest10b  32223  dibopelvalN  35450  dibopelval2  35452  climrec  38670