Theorem elsn2 4617

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 4616	. 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 3436  {csn 4577

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 4578

This theorem is referenced by:  fparlem1  8045  fparlem2  8046  el1o  8413  fin1a2lem11  10304  fin1a2lem12  10305  elnn0  12386  elxnn0  12459  elfzp1  13477  fsumss  15632  fprodss  15855  elhoma  17939  rnglidl0  21136  islpidl  21232  zrhrhmb  21417  rest0  23054  qustgphaus  24008  taylfval  26264  eqscut3  27735  elch0  31198  atoml2i  32327  prmidl0  33387  bj-eltag  36955  bj-rest10b  37067  dibopelvalN  41126  dibopelval2  41128  aks4d1p1p4  42048  climrec  45588