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Theorem sneqr 4311
 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
StepHypRef Expression
1 sneqr.1 . 2 𝐴 ∈ V
2 sneqrg 4310 . 2 (𝐴 ∈ V → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
31, 2ax-mp 5 1 ({𝐴} = {𝐵} → 𝐴 = 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sn 4126 This theorem is referenced by:  snsssn  4312  sneqrgOLD  4313  opth1  4870  propeqop  4895  opthwiener  4901  funsndifnop  6321  canth2  7998  axcc2lem  9141  hashge3el3dif  13122  dis2ndc  21073  axlowdim1  25639  bj-snsetex  32144  poimirlem13  32592  poimirlem14  32593  wopprc  36615  hoidmv1le  39484
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