Theorem elsng 4641

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	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem elsng

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2737	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
2		df-sn 4628	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
3	1, 2	elab2g 3669	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-sn 4628

This theorem is referenced by:  elsn  4642  elsni  4644  snidg  4661  eltpg  4688  eldifsn  4789  sneqrg  4839  elsucg  6429  ltxr  13091  elfzp12  13576  fprodn0f  15931  lcmfunsnlem2  16573  ramcl  16958  initoeu2lem1  17960  pmtrdifellem4  19340  logbmpt  26273  2lgslem2  26878  elunsn  31728  xrge0tsmsbi  32188  dimkerim  32657  elzrhunit  32897  esumrnmpt2  33004  plymulx  33497  bj-projval  35815  bj-elsn12g  35879  bj-elsnb  35880  bj-snmoore  35932  bj-elsn0  35974  eldmressnALTV  37078  brressn  37249  aks4d1p6  40884  sticksstones11  40910  metakunt20  40942  reclimc  44304  itgsincmulx  44625  dirkercncflem2  44755  dirkercncflem4  44757  fourierdlem53  44810  fourierdlem58  44815  fourierdlem60  44817  fourierdlem61  44818  fourierdlem62  44819  fourierdlem76  44833  fourierdlem101  44858  elaa2  44885  etransc  44934  qndenserrnbl  44946  sge0tsms  45031  el1fzopredsuc  45968  mndtcob  47610