Theorem sneqr 4806

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 4805	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463  {csn 4591

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-sn 4592

This theorem is referenced by:  snsssn  4807  mosneq  4808  opth1  5455  propeqop  5488  opthwiener  5495  funsndifnop  7146  canth2  9114  axcc2lem  10416  hashge3el3dif  14520  dis2ndc  23582  axlowdim1  29246  selvply1rhmlema  33849  selvply1rhmlem1  33851  esplyfval1  33904  mh-inf3sn  36938  bj-snsetex  37483  poimirlem13  38167  poimirlem14  38168  wopprc  43642  snen1g  44135  mnuprdlem2  44868  hoidmv1le  47193  fsetsnf1  47671