Theorem elsn2 4646

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 4645	. 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 3464  {csn 4606

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-sn 4607

This theorem is referenced by:  fparlem1  8116  fparlem2  8117  el1o  8512  fin1a2lem11  10429  fin1a2lem12  10430  elnn0  12508  elxnn0  12581  elfzp1  13596  fsumss  15746  fprodss  15969  elhoma  18050  rnglidl0  21195  islpidl  21291  zrhrhmb  21476  rest0  23112  qustgphaus  24066  taylfval  26323  elch0  31240  atoml2i  32369  prmidl0  33470  bj-eltag  37000  bj-rest10b  37112  dibopelvalN  41167  dibopelval2  41169  aks4d1p1p4  42089  climrec  45599