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Theorem elsng 4139
 Description: There is exactly one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
elsng (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsng
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2614 . 2 (𝑥 = 𝐴 → (𝑥 = 𝐵𝐴 = 𝐵))
2 df-sn 4126 . 2 {𝐵} = {𝑥𝑥 = 𝐵}
31, 2elab2g 3322 1 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126 This theorem is referenced by:  elsn  4140  elsni  4142  snidg  4153  nelsnOLD  4160  eltpg  4174  eldifsn  4260  sneqrg  4310  elsucg  5709  ltxr  11825  elfzp12  12288  fprodn0f  14561  lcmfunsnlem2  15191  ramcl  15571  initoeu2lem1  16487  pmtrdifellem4  17722  logbmpt  24326  2lgslem2  24920  nbcusgra  25992  frgrancvvdeqlem1  26557  xrge0tsmsbi  29117  elzrhunit  29351  elzdif0  29352  esumrnmpt2  29457  plymulx  29951  bj-projval  32177  reclimc  38720  itgsincmulx  38866  dirkercncflem2  38997  dirkercncflem4  38999  fourierdlem53  39052  fourierdlem58  39057  fourierdlem60  39059  fourierdlem61  39060  fourierdlem62  39061  fourierdlem76  39075  fourierdlem101  39100  elaa2  39127  etransc  39176  qndenserrnbl  39191  sge0tsms  39273  el1fzopredsuc  39948  frgrncvvdeqlem1  41469
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