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 2145  Vcvv 3450  {csn 4584

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-sn 4585

This theorem is used by:  fparlem1  8110  fparlem2  8111  el1o  8485  fin1a2lem11  10415  fin1a2lem12  10416  elnn0  12533  elxnn0  12606  elfzp1  13632  fsumss  15814  fprodss  16038  elhoma  18124  rnglidl0  21421  prmidl0  21544  islpidl  21559  zrhrhmb  21726  rest0  23397  qustgphaus  24352  taylfval  26598  eqcuts3  28072  elch0  31738  atoml2i  32867  bj-eltag  37724  bj-rest10b  37842  dibopelvalN  42019  dibopelval2  42021  aks4d1p1p4  42940  climrec  46436  tmachlem-agreesn  47778