Theorem sneqr 4784

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 4783	. 2 ⊢ (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
3	1, 2	ax-mp 5	1 ⊢ ({𝐴} = {𝐵} → 𝐴 = 𝐵)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  {csn 4568

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sn 4569

This theorem is referenced by:  snsssn  4785  mosneq  4786  opth1  5423  propeqop  5455  opthwiener  5462  funsndifnop  7098  canth2  9061  axcc2lem  10349  hashge3el3dif  14440  dis2ndc  23435  axlowdim1  29042  esplyfval1  33732  mh-inf3sn  36740  bj-snsetex  37286  poimirlem13  37968  poimirlem14  37969  wopprc  43476  snen1g  43969  mnuprdlem2  44718  hoidmv1le  47040  fsetsnf1  47512