Theorem elsn2 4625

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 4624	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  Vcvv 3444  {csn 4585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sn 4586

This theorem is referenced by:  fparlem1  8068  fparlem2  8069  el1o  8436  fin1a2lem11  10339  fin1a2lem12  10340  elnn0  12420  elxnn0  12493  elfzp1  13511  fsumss  15667  fprodss  15890  elhoma  17970  rnglidl0  21115  islpidl  21211  zrhrhmb  21396  rest0  23032  qustgphaus  23986  taylfval  26242  elch0  31156  atoml2i  32285  prmidl0  33394  bj-eltag  36938  bj-rest10b  37050  dibopelvalN  41110  dibopelval2  41112  aks4d1p1p4  42032  climrec  45574