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Theorem sneqr 4302
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 4301 . 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 1474  wcel 1975  Vcvv 3168  {csn 4120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-v 3170  df-sn 4121
This theorem is referenced by:  snsssn  4303  sneqrgOLD  4304  opth1  4860  opthwiener  4888  canth2  7971  axcc2lem  9114  hashge3el3dif  13068  dis2ndc  21011  axlowdim1  25553  bj-snsetex  31943  poimirlem13  32391  poimirlem14  32392  wopprc  36414  hoidmv1le  39284  propeqop  40122  funsndifnop  40144
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