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Theorem sneqr 4197
 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
Assertion
Ref Expression
sneqr

Proof of Theorem sneqr
StepHypRef Expression
1 sneqr.1 . . . 4
21snid 4057 . . 3
3 eleq2 2530 . . 3
42, 3mpbii 211 . 2
51elsnc 4053 . 2
64, 5sylib 196 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818   cvv 3109  {csn 4029 This theorem is referenced by:  snsssn  4198  sneqrg  4199  opth1  4725  opthwiener  4754  canth2  7690  axcc2lem  8837  hashge3el3dif  12524  dis2ndc  19961  axlowdim1  24262  wopprc  30972  bj-snsetex  34521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sn 4030
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