Theorem elsn2 4626

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 4625	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-sn 4589

This theorem is used by:  fparlem1  8147  fparlem2  8148  el1o  8528  fin1a2lem11  10512  fin1a2lem12  10513  elnn0  12632  elxnn0  12705  elfzp1  13732  fsumss  15917  fprodss  16141  elhoma  18233  rnglidl0  21536  prmidl0  21661  islpidl  21676  zrhrhmb  21843  rest0  23514  qustgphaus  24469  taylfval  26713  eqcuts3  28217  elch0  31883  atoml2i  33012  bj-eltag  38067  bj-rest10b  38200  dibopelvalN  42391  dibopelval2  42393  aks4d1p1p4  43312  climrec  46818  tmachlem-agreesn  48160  parhomel  50947  parhomcl2  50948