Theorem elsn2 4610

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 4609	. 2 ⊢ (𝐵 ∈ V → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sn 4569

This theorem is referenced by:  fparlem1  8056  fparlem2  8057  el1o  8424  fin1a2lem11  10326  fin1a2lem12  10327  elnn0  12433  elxnn0  12506  elfzp1  13522  fsumss  15681  fprodss  15907  elhoma  17993  rnglidl0  21222  islpidl  21318  zrhrhmb  21503  rest0  23147  qustgphaus  24101  taylfval  26338  eqcuts3  27813  elch0  31343  atoml2i  32472  prmidl0  33528  bj-eltag  37303  bj-rest10b  37420  dibopelvalN  41606  dibopelval2  41608  aks4d1p1p4  42527  climrec  46054