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Theorem sneqr 4362
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 4361 . 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 1481  wcel 1988  Vcvv 3195  {csn 4168
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-sn 4169
This theorem is referenced by:  snsssn  4363  sneqrgOLD  4364  opth1  4934  propeqop  4960  opthwiener  4966  funsndifnop  6401  canth2  8098  axcc2lem  9243  hashge3el3dif  13251  dis2ndc  21244  axlowdim1  25820  bj-snsetex  32926  poimirlem13  33393  poimirlem14  33394  wopprc  37416  hoidmv1le  40571
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