Theorem elsng 4191

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 2626	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐵 ↔ 𝐴 = 𝐵))
2		df-sn 4178	. 2 ⊢ {𝐵} = {𝑥 ∣ 𝑥 = 𝐵}
3	1, 2	elab2g 3353	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  elsn  4192  elsni  4194  snidg  4206  nelsnOLD  4213  eltpg  4227  eldifsn  4317  sneqrg  4370  elsucg  5792  ltxr  11949  elfzp12  12419  fprodn0f  14722  lcmfunsnlem2  15353  ramcl  15733  initoeu2lem1  16664  pmtrdifellem4  17899  logbmpt  24526  2lgslem2  25120  xrge0tsmsbi  29786  elzrhunit  30023  elzdif0  30024  esumrnmpt2  30130  plymulx  30625  bj-projval  32984  bj-snmoore  33068  reclimc  39885  itgsincmulx  40190  dirkercncflem2  40321  dirkercncflem4  40323  fourierdlem53  40376  fourierdlem58  40381  fourierdlem60  40383  fourierdlem61  40384  fourierdlem62  40385  fourierdlem76  40399  fourierdlem101  40424  elaa2  40451  etransc  40500  qndenserrnbl  40515  sge0tsms  40597  el1fzopredsuc  41335