Theorem sneqr 4805

Description: If the singletons of two sets are equal, the two sets are equal. Part of Exercise 4 of [TakeutiZaring] p. 15. (Contributed by NM, 27-Aug-1993.)

Hypothesis

Ref	Expression
sneqr.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
sneqr	⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Proof of Theorem sneqr

Step	Hyp	Ref	Expression
1		sneqr.1	. 2 ⊢ 𝐴 ∈ V
2		sneqrg 4804	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Vcvv 3455  {csn 4589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-sn 4590

This theorem is referenced by:  snsssn  4806  mosneq  4807  opth1  5457  propeqop  5490  opthwiener  5497  funsndifnop  7148  canth2  9114  axcc2lem  10415  hashge3el3dif  14520  dis2ndc  23617  axlowdim1  29309  selvply1rhmlema  33908  selvply1rhmlem1  33910  esplyfval1  33963  mh-inf3sn  37053  bj-snsetex  37599  poimirlem13  38284  poimirlem14  38285  wopprc  43757  snen1g  44250  mnuprdlem2  44983  hoidmv1le  47308  fsetsnf1  47789